ASTROPHYSICAL MONOGRAPHS
Sponsored by
THE ASTROPHYSICAL JOURNAL
Edited by
HENRY Ci. GALE FREDERICK H. SEARES
Ryerson Physical Laboratory Mount IVilson Observatory
oj the Unioersity oj Chicago of the Carnegie Institution oj Washington
o T'ro s'rRUVE
Yerkes Observatory
oJ the University oJ Chicagt
AN IN'rRODUCTION TO THE STUDY
OF STELLAR STRUCTURE
THE UNIVERSITY OF CHICAGO PRESS, CHICAGO
TOE BAKER & TAYLOR COMPANY, NEW YORK; THE CAMBRIDGE UNIVERSITY
PRESS, LONDON; THE MARUZEN-KABUSHIKI- KAISHA, TOKYO, OSAKA,
KYOTO, FUKUOKA, SENDAI; THE COMMERCIAL PRESS, LIMITED, SHANGHAI
AN INTRODUCTION
TO THE STUDY OF STELLAR
STRUCTURE
Bv S. CHANDRASEKHAR
Yn'kes Observatory
ri lK UNIVKRSI IT OK CHICAGO PRKSS
CHICACJO ■ ILLINOIS
COPYRIGHT 1939 BY THE UNIVERSITY OF CHICAGO. ALL RIGHTS
RESERVED. PUBLISHED JANUARY I939. COMPOSED AND PRINTED
BY THE UNIVERSITY OF CHICAGO PRESS, CHICAGO, ILLINOIS, U.S.A.
PREFACE
The present volume forms the second in the series of the ^‘Astro-
physical Monographs.” The plan and scope of this book are set
forth in the introductory chapter, and there remains only the pleas-
ant task of thanking those who have helped me. I am under obliga-
tion to Mr. B. Strdmgren for many valuable discussions during the
writing of the monograph and also for allowing me to incorporate in
chapters vi and vii some of his unpublished investigations. In the
same way, Mr. G. P. Kuipcr has allowed me to use the results of his
study on the empirical mass-luminosity relation before publication.
I am also deeply grateful to Mr. W. W. Morgan for reading the
whole book both in manuscript and in proof. It is also a pleasure to
record the generous encouragement I have received from Mr. 0.
Struve.
Finally, I wish to express my very grateful appreciation of the
unfailing courtesy and consideration which the oHlcials of the Uni-
versity of Chicago Press have shown me during the printing of this
volume.
S. C.
VkKKK.S OMSKRYA'I'ORY
I)tTcnil)er ic^iS
vii
TABLE OF CONTENTS
page
Introduction i
CHAPTER
I. The Laws OF Thermodynamics ii
II. Physical Principles 38
ill. Integral 'Fheorems on the Equilibrium of a Star .... 61
IV. Polytropic and Isothermal Gas Spheres 84
V. 'I'liE 'Fiieory of Radiation and the Equations of Equilibrium 183
VI. Gaseous Stars 216
VI I . Stromgren’s Interpretation of the Hertzsprung-Russell
Diagram 249
VI II. Stellar Envelopes and the Central Condensation of Stars 292
IX. Stellar Models 322
X. 'Fhe Quan'itim Statistics 357
XI. Degenerate Stellar Configurations and the Theory of
White D wakes 412
XI 1 . Stellar Ener(;y 453
ai'I'Knuixks
I. I’liYSK’AL AND As'I'ronomicai. Constants 487
II . 'Fiie Masses <»E the Lkhit Atoms 488
III. M'iie Masses, Luminosities, and Radii of the Stars; Derived
IIydhoc.en C'ontents; C'icntral Densities; and Central Tem-
peratures 4^9
IV. 'Fables OF the Wiutic-Dware Functions 491
General Index 5^5
IX
INTRODUCTION
In this monograph an attempt is made to develop the theory of
stellar structure from a consistent point of view and, as far as possi-
ble, rigorously. This and considerations of space have placed a some-
what severe restriction on the problems that are to come under re-
view, while requiring at the same time a detailed treatment of Other
aspects of the subject. Thus, on the physical side, questions requir-
ing the application of relatively advanced methods of statistical me-
chanics have to be avoided, while, on the astronomical side, ques-
tions concerned with problems of the type of stellar rotation and
stellar variability or stability have had to be entirely omitted. This
may seem a drawback, but, on the other hand, there is more space
to develop the fundamentals with which the reader should be thor-
oughly familiar. In this introduction we shall make some comments
on the type of problems with which we shall be mainly concerned
and then outline the plan and scope of the monograph.
As we have already indicated, we shall restrict ourselves to the
consideration of stars which are in equilibrium and which arc in a
steady state. Such an eciuilibrium configuration can be character-
ized by three i)arameters: its mass, M; its radius, R; and its luminos-
ity, L (L being defined as the amount of radiant energy, expressed in
ergs, radiated by the star ])er second to the space outside). It is
beyond the sco])e of the monograi)h to discuss how the values of
these ])arameters for individual stars arc determined in practice.
We shall assume, however, that we do have sets of values of these
quantities for a number of stars. Stellar structure deals with these
results of observational astronomy.
Our first problem, then, is to })resent the observational material
in some form suitable for further discussion. There are two plots
which we shall find useful: (a) the mass-luminosity diagram, and
ih) the mass-radius diagram. In diagram {(/) it is customary to plot
log M {M ex])ressed in solar units) against, essentially, 2.5 log L
(in i)ractice, the absolute bolometric magnitude). In diagram {b) we
2
STUDY OF STELLAR STRUCTURE
plot log M against log R {R expressed in solar units). In Figures i
and 2 we have collected together the results of observations; the
material presented has been provided by Dr. G. P. Kuiper.
The ultimate objects of studies in stellar structure are the follow-
ing:
1. To derive the complete march of the physical variables (the
density, p; the temperature, T; etc.), on the one hand, and the vari-
ation of the chemical composition (the relative abundances of the
different elements), on the other, throughout the entire configura-
tion.
2. To describe quantitatively the kind of steady state (radiative,
convective, etc.) that exists, eventually as a function of the radius
vector r.
3. To specify the fundamental physical processes that are re-
sponsible for the setting-up of the steady states described under (2).
4. To evaluate quantitatively the irreversible processes that must
be taking place which should be responsible for the continual loss of
energy at the rate L by a star.
It is clear that complete and entirely satisfactory answers to all
the foregoing problems require detailed information about physical
phenomena which we do not have at the present time; even if we
possessed this information, we should be faced with a mathematical
problem of a very high order of complexity. From one point of view
the most serious lack of information (at least until recently) con-
cerns the nature of the physical processes involved under (4) above.
The question now arises as to how we can formulate, at least
provisionally, the fundamental problem of stellar structure the solu-
tion of which wiU not only be of value but will also enable us to make
substantial progress toward the solution of the complete problem.
In other words, we need to formulate a somewhat restricted prob-
lem of stellar structure. The problem we shall consider is: Can we
establish some relation between all three parameters, L, M, and R?
That we can hope to make some progress toward the solution of
this problem can be seen in the following way. When we observe a
star, we see that in a prescribed spherical volume of radius R an
amount of material of total mass M is inclosed ; we also know that
through this mass there occurs a continual streaming-out of a cer-
nes;
Fig. 2. — The mass-radius diagram. Solid dots: visual binaries; open circles: spectroscopic binaries;
vertical crosses: the stars in the Hyades; diagonal crosses: Ihe Trumpler stars, squares:* white dwarfs.
INTRODUCTION
S
tain mean flux of radiant energy specified by the luminosity, L.
By hypothesis the star is in a steady state. The question we can
then ask is: ‘‘How is it that a certain specified march of the net
flux of radiant energy is able to support (against the gravitational
attraction) an amount of mass equal to M inside a spherical volume
of precisely the radius, R?'' It will be noticed that some uncertainty
has already been introduced. The luminosity, L, specifies the net
flux of energy given by at the boundary of the star; we can,
of course, take this as an index of a certain average flux that exists
in the interior, but the solution of the mathematical problem of
equilibrium would require a knowledge of the complete march of
the function L{r) and not merely a certain unspecified average de-
pending on L. It is precisely for this reason that progress toward the
solution of the restricted problem is made by means of the study of
stellar models.
From the observed L and M we infer that each gram of the stellar
material liberates on the average an amount of energy, s = L/M,
It may be safely assumed that €(r) — the rate of liberation of energy
per gram of the material at the point r — is zero in the outer parts
of the star where the physical conditions arc relatively “mild,*^ so
that /Ar) = L in the outer parts of the star (these parts constitute
the stellar envelope studied in chapter viii). Presumably, L{r) de-
creases inward in such a way that c(f) = L(r)/M(r) tends to some
finite value as r o, with or without a maximum for r > o (in
the latter case €(o) will be the maximum value of the function €(r)).
1 wo obvious limiting cases suggest themselves: {a) €(r) = e =
Constant, and (h) e(r) = o, r 4- o. The former case corresponds
to a uniform distribution of the energy sources, while the latter
leads us to the case where all the energy sources arc concentrated
at the center (this is the “point-source modeP^).
We can investigate these two limiting cases as well as other, “in-
termediate,” stellar models. After studying such models we attempt
to abstract from the ensemble of the results thus obtained features
which can be regarded as common to all the models. It would be
safe to conclude that such common features must have some coun-
ter])art in nature; this is the manner in which progress has been
made. One rather unexpected feature introduces an essential simpli-
6
STUDY OF STELLAR STRUCTURE
fication. We shall discuss the origin of this simplification later
(chaps, ii, vii, and viii) ; but it may be stated here that it follows
from very general considerations that the majority of the normal
stars, such as the sun and Capella, are gaseous and that radiation
pressure as a factor in the equation of the hydrostatic equilibrium
can be neglected (though, of course, it is important in determining
the temperature gradient set up). This last circumstance in turn re-
veals another unexpected feature : the form of the relation between
L, M, and R is independent of the stellar model considered. We shall
not go farther into the consideration of these matters, but enough
has been said to show that progress toward the solution of the re-
stricted problem is in fact possible.
There is one other matter of importance to which we shall draw
attention: we cannot assume beforehand that the chemical compo-
sition of all the stars is the same. Actually, under stellar conditions
matter is generally so highly ionized that, as we shall see in greater
detail in chapter vii, the uncertainty in the chemical composition
is essentially due to the uncertainty in the abundance of the two
lightest elements, namely, hydrogen and helium. The abundance of
the lightest elements has then to be considered as a fresh parameter
in the discussion. We can thus summarize by saying that our funda-
mental problem is to seek a theoretical relation of the kind
F\L, M, R, abundance of hydrogen and helium] = o . (1)
Our main object, then, is to describe the theory and the methods
that have been developed toward this end.
We shall now proceed to outline the general plan :
The monograph divides itself into two distinct parts: the “classi-
cal” (chaps, i-iv) and the “modern” (chaps, v-xii). Furthermore,
of the twelve chapters, two (chaps, i and x) deal essentially with
physical theories (the laws of thermodynamics and quantum statis-
tics, respectively), and chapter v deals also with a physical theory
(the formal theory of radiation) presented, however, from an astro-
physical angle. The last, chapter xii, on stellar energy, is on a plane
different from the rest in that it summarizes the recent work on some
INTRODUCTION
7
of the most thorny problems of the subject. In greater detail the
contents of each of the chapters are as follows:
Chapter i . — The laws of thermodynamics are here presented fol-
lowing Carath6odory’s axiomatic standpoint. The reasons for includ-
ing this chapter are twofold: first, there exists no treatise in English
which gives Carath6odory’s theory; and second, in the writer’s view
Caratheodory’s theory is not merely an alternative, but elegant, ap-
proach to thermodynamics but is the only physically correct approach
to the second law. Incidentally, the logical rigor and the beauty of
Caratheodory’s theory may be regarded as an example of the stand-
ard of perfection which should be demanded eventually of any physi-
cal theory, including the theory of stellar structure.
Chapter ii . — In this chapter we consider a number of physical
theorems, the adiabatic and the polytropic laws, the virial theorem,
homologous transformations, etc., and some immediate applications
to the general theory of stellar structure.
Cha pier Hi . — Here we attempt to go as far as possible with our
problem without any special assumptions except that the stars are
in hydrostatic equilibrium. It is made clear in this chapter that no
special assumptions arc required to derive the orders of magnitude
of the most important physical quantities which describe the struc-
ture of a star.
Chapter iv.- This chapter presents what is perhaps the most im-
portant contribution which “stellar structure’’ has made to applied
mathematics. It represents, largely, the work of the great pioneers-
Ritter, Emdcn, and Kelvin. As Schwarzschild has said, the theory
of jxjlytropcs is a beautiful example of the flowering of a complete
mathematical theory out of a physical problem. The bibliographical
note for this chapter has been made rather extensive, as there ap-
pears to be, at the present time, a great deal of confusion with re-
gard to the historical developments of the subject. It may be stated
further that no fundamentally new contribution has been made to
the subject since the publication of Emden’s book (1907).
Chapter n. Here the formal theory of radiation is presented and
the equations of radiative equilibrium are derived, 'i'he number of
final results obtained is small, but the amount of formal develop-
8
STUDY OF STELLAR STRUCTURE
merits required is rather considerable; for accuracy and precision
they cannot, however, be avoided.
Chapter vi. — In this chapter "gaseous stars” are considered. Some
general theorems for stars in radiative equilibrium are obtained, and
a fundamental formula — the luminosity formula — ^is derived. This
last is made the starting-point of the whole discussion.
Chapter mi. — ^Here the general theory (which leads to a definite
relation of the type; I) is used to derive from the observational ma-
terial the abundances of the lightest elements for individual stars,
and an attempt is made to draw some general conclusions. The em-
phasis thus laid on Stromgren’s work on the varying abundance of
the lightest elements from star to star is one of the more important
features of the monograph. In discussing this theory it is important
to realize that the theory of the stellar absorption coefficient and of
the mean molecular weight (which are also described in this chapter)
have been developed as accurately as is necessary for the purposes
at hand.
Chapter mu. — ^In this chapter the theory of stellar envelopes is
made an independent starting-point for the theory of gaseous stars.
This serves partly to confirm the results described in chapter vii and
partly to go beyond the range of that theory.
Chapter ix. — Some further stellar models are considered in this
chapter which partly confirm the results of chapter vi and partly
extend them.
Chapter x. — rather detailed account of the Gibbs statistical
mechanics (the quantum mechanical version) is given in this chaj)-
ter. In view of the astrophysical applications the theory is developed
to take account of the relativistic effects from the outset.
Chapter xi. — ^The theory of degeneracy developed in chapter x is
here applied to elucidate the structure of the white dwarfs. For the
white dwarfs the structure depends, to a good approximation, only
on M, R, and the abundance of the lightest elements. This result
arises essentially from the circumstance of the white dwarfs being
highly “underluminous.”
Chapter xii. — In this chapter some general trends in the current
investigations on the problem of stellar energy are outlined. There
INTRODUCTION
9
are, as yet, no very definitive results to report, but some general ideas
which are likely to prove fruitful in the future developments of the
subject are considered.
The foregoing brief comments on the contents of each of the chap-
ters may be supplemented by their introductory paragraphs, where
more specific statements of the problems considered are made.
In concluding this brief outline, of the monograph, it should be
emphasized again that the particular arrangement of the subject
matter has arisen in the attempt to present the subject from a uni-
fied standpoint. This, in turn, has required a somewhat detailed
treatment of certain aspects of the subject which may not appear
to deserve that prominence. However, in the opinion of the writer
the general standpoint taken appears to be the only fruitful one
under the present limitations of our knowledge.
Finally, in the actual developments an attempt has been made to
give the full details, both of mathematical derivations and of physi-
cal theories, as far as this has proved feasible. This method may in-
volve the disadvantage that there is a danger of the reader losing
the general perspective in the details of the solution of a mathemati-
cal problem or in the arguments of a physical theory. It will there-
fore be advantageous — even though it may not be strictly necessary
if the reader acquires during the study of the monograph some
familiarity with the general results. For an attractive account, which
in several respects runs parallel to the more detailed treatment of
the monograph, reference may be made to B. Strdmgren, ^‘Die Theo-
rie (les Sterninnern und die Entwicklung der Sterne,’’ Rrgebnisse der
Exakien NaturwissenschaJtcnj i6, 465, 1937.
We shall now consider two ‘technical” matters concerning the
monograph.
I. Bibliographical -With regard to references to the litera-
ture, it was decided to resist the temptation of making it a rule to
give running references in the text; actually only a very few refer-
ences are made. This has resulted in a more continuous arrangement
of the arguments than would otherwise have been possible. How-
ever, at the end of each chapter bibliographical notes are appended
lO
STUDY OF STELLAR STRUCTURE
in which specific references to each particular section are made. It
should, moreover, be stated that it has not been the intention even
to attempt to give a complete list of references; only those investiga-
tions are quoted which have been incorporated in the text or the
results of which further amplify the points concerned.
2. The numbering of the sections and the equations . — ^The equations
and the sections in the different chapters have been numbered sepa-
rately. References to equations or sections (§) in the same chapter
are made by giving the appropriate numbers, e.g., equation (37)
or § 10; references to equations or sections in a different chapter are
distinguished by giving the chapter number as a Roman numeral,
e.g., equation (37), v (or more simply as (37), v), or § 10, v.
CHAPTER I
THE LAWS OF THERMODYNAMICS
In this chapter we shall be concerned mainly with the first and
the second laws of thermodynamics. In our presentation of the fun-
damental principles of thermodynamics we shall follow Caratheo-
dory’s axiomatic point of view. This axiomatic presentation of the
laws of thermodynamics has the advantage of reducing the number
of new undefinables to a minimum and achieves at the same time
the maximum logical simplicity. Since a proper appreciation of the
meaning and content of the laws of thermodynamics is necessary
for the developments in the succeeding chapters, we shall accord-
ingly develop the fundamental ideas ab initio,
I. We shall consider only the simplest of thermodynamical sys-
tems, namely, those composed of chemically noninteracting mixtures
of gases and liciiiids. We shall assume that the elementary notions
concerning mass, force, pressure, work, and volume are familiar; we
shall, however, deline accurately the purely thermal notions, such as
“lemperature,’’ ‘^quantity of heat,” etc.
In the purely mechanical discussions of the equilibrium of a body
- as, for instance, in hydrodynamics the inner state of a fluid of
known mass is determined when we know its specific volume, F, the
volume per unit mass of the Iluid. But this is not generally true, as
we can alter the pressure exerted by a gas without altering its
s])ecific volume, V , For this purpose it is necessary to consider physi-
cal processes which are associated with '^heating.” In thermodynam-
ics such physical situations are realized, and we introduce both the
pri'ssure, /?, and the volume, F, as independent variables. Thus, F
and p specify com])letely the inner state of a system.
We assume that individual systems can be isolated from the out-
side world by means of inclosures, or that two parts of a given system
can be se])arated by walls. Though we shall not include these in-
closures or walls as a ])art of the thermodynamical system, we shall
yet have to make certain sixrilic ideal requirements for these parti-
tions. We shall have to consider two types of such partitions.
12
STUDY OF STELLAR STRUCTURE
a) Adiabatic inclosures . — a body is indosed in an adiabatic in-
closure and if it is in equilibrium, then, in the absence of external
fields of forces, the only way in which we can change the inner state
of the body is by means of actual displacements of at least some
finite part of the walls of the inclosure. If we assume the notion of
heat, this means that the only way in which we can change the
inner state of a body in an adiabatic indosure is by doing external
work, and that, furthermore, the walls of the indosure are opaque
to the communication of heat.
h) Diathermic partitions . — ^If two bodies are indosed in an adia-
batic indosure but are mutually separated by a diathermic wall,
then a certain definite relation between the four parameters pi, F,;
pt, Vx (defining the state of the two bodies, respectively) must exist
in order that there may be equilibrium; the relation depends on the
nature of the two bodies only. Thus, we must have
P(.Pi> Fj, pi, Fa) = o . (i)
We shall say that two bodies are in “thermal contact” if they are
both inclosed in the same adiabatic indosure but are separated by
a diathermic wall. Equation (i) then expresses the condition for
thermal equilibrium.
Thus, it is empirically found that, if two perfect gases are in ther-
mal contact, we always have
piV , — piVi = o .
2 . Empirical temperature . — Experience shows the following char-
acteristic of thermal equilibrium. If {p,, F.), (/>„ F,), {p,, F,), and
y 0 define two distinct states of two different systems (not nec-
essarily those of two different bodies) and if both {p„ F,) and
{px, Fj) are in thermal equilibrium with {p,, F,), and if, further,
{Pi, y i) is in thermal equilibrium with {px, F^), then it is always true
that {Px, Fj) will be in thermal equilibrium with {pi, V^. This sim-
ply means that, if two bodies are separately in thermal equilibrium
with a third body, then the two original bodies, if brought into ther-
mal contact, would also be in thermal equilibrium. By equation (i),
THE LAWS OF THERMODYNAMICS
13
which specifies the condition for thermal equilibrium, the foregoing
means that the equations
Hh, h, V.) = o, F(p2, p., F,) = o ,
HP., V„ p., F,) = o , /
imply the validity of
P(p>, V2, p2, F,) = o . (3)
But this is then, and only then, possible if the relation F{p, V,P,Y)
= o has the form
I{p, V) - 1 (P, F) = 0 . ( 4 )
In (4) t and I arc not uniquely determined, for the condition of
equilibrium, (4), can also be written as
T\i{p, F)i = mp, F)] , (4')
where 7 ’(.v) can be any arbitrary function in x.
Of all the possible forms which the condition of equilibrium can
take, let us choose arbitrarily one particular form and write it in
the fomi (4). 'I’he values lip, V) and lip, V) deline on an arbitrary
scale the empirical Icmpcralurc of the two bodies; if the two bodies
are in thermal contact and are in equilibrium, then we should always
have the eciualily of the eminrical temperatures. If
I = tip, r), I = lip, F) , (5)
then in equilibrium
1 = 1 . ( 6 )
'I'he ec|uations (5) define in the ip, F) and in the ip, V) planes, re-
spectively, a one-parametric family of curves which are called “iso-
thermals.” 'I'he equations (5) are calleii the “eciuations of state.”
If the empirical temperature scale is once selected and dehned,
then we can always choose any two of the three variables p, V , and I
as the independent variables defining the state of a system. In the
14
STUDY OF STELLAR STRUCTURE
same way two arbitrary functions of the physical variables F,
and t would also sufiice to specify a state of the system.
3. The First Law of Thermodynamics experiments of Joule
establish the following circumstance :
In order to bring a body {or a system of bodies) from a prescribed
initial state to another prescribed final state adiabatically, then the
same constant amount of mechanical work {or an equivalent electrical
work), which is independent of how the change is carried out and which
depends only on the prescribed initial and final states, has to be done.
Let the initial state be specified by ^0, Vo, , and the final
state by pi, Vi, ... . Let the work done to carry out the change of
state adiabatically be W. Then, according to the first law, if we keep
the initial state fixed, W depends only on the final state. We can
therefore write
TF=C/-£/o, (7)
where C/ is a function of the parameters determining the state of the
system — p and V, if there is only one body — and Uo is its value in
the initial state. U, thus defined, is called the ^'internal energy’" of
the system.
If we define our unit of heat as the mechanical work (expressed
in ergs) required to change the (empirical) temperature, t, of water
of unit volume (at constant volume) between two definite values,
then we obtain the so called ^‘mechanical equivalent of heat.”
4. Quantity of heat . — Suppose that we know the internal energy
as a function of the physical parameters from a series of calorimetric
experiments, as, for instance, Joule’s experiments. Suppose, now,
that in some given arbitrary nonadiabatic process the internal
energy of a system changes by (t/* — Uo); further, let W be the
amount of work done on the system. Then we say that a quantity
Q of heat, where
{U -Uo)-W, (8)
has been supplied to the system.
We see that the notion of the quantity of heat has no independent
meaning apart from the First Law of Thermodynamics. {U — Uo)
THE LAWS OF THERMODYNAMICS 15
is a physical quantity which can be determined experimentally,
while the notion of Q is a derived one.
5. The internal energy of a system of bodies. — If two or more bodies
arc isolated from each other adiabatically, then by definition the
energy of the system is equal to the sum of the energies of the indi-
vidual bodies :
i/ = f/i + C/a . (9)
In general, when the two bodies are brought into contact, the energy
is not additive; it is easy to see, however, that the deviation must be
proportional to the common surface area of the bodies, and hence, for
large volumes the deviations from the additive law can be neglected.
6. Stillionary and qiiasi-statical processes. — In the formulation of
the first law we assumed that the work done can in principle be meas-
ured. But to evaluate the work done during a given process we need
an api)aratus to register continuously the forces exerted on, and the
(lisidaeements of, the walls of the inclosure, for the work done is
simply the integral over the product of the force and the displace-
ment. In practice this limits us to only two essentially distinct pro-
cedures for which wc can measure the work done. These are:
a) Slalionary procc.sscs. -For example, as in Joule's experiments,
there is a stirrer which rotates in the fluid at a constant rate. This
would give rise to a stationary system of currents in which the stirrer
ex])eriences a constant friction. If we neglect the relatively small
accc‘leration in the l)eginning and the end of the interval during
which the stirrer rotates, then the work done is simply the product
of the torque limes the rate of working of the stirrer.
/>) Quasi-slaliral processes. We conduct the process infinitely
slowly, so that we can regard the state of the system at any given
moment as one of eriuilibrium. We refer to such processes as ^'quasi-
statieal j)rocesses.” d'hey are generally referred to as “reversible
I)r<)ci'ss(‘s” hei'ause, in general, cjuasi-statical processes can be con-
ductt‘(l in the riwersc sense. We shall refer to a process as “nonstati-
eal” if it is not quasi-statical.
7. fnfinilesinial qiiasi-slalical adiabatic cluinges. - wc have a
body im losed in an adiabatic inclosure, and if we do an infinitesimal
amount of mechanical work, dW (by displacing the walls of the in-
i6
STUDY OF STELLAR STRUCTURE
closure), carried out quasi-statically, then we say that we have car-
ried out an ‘ 'infinitesimal quasi-statical adiabatic change.” If dur-
ing such an infinitesimal quasi-statical adiabatic change the change
in volume amounts to dV^ then clearly
dW = -pdV , (lo)
where p is the equilibrium pressure. Then, according to the first law,
dQ = dU + pdV = o, (ii)
For a system of two bodies which are both inclosed in the same
adiabatic inclosure but which are separated from one another by
means of a diathermic wall, we have, since both Q and U are addi-
tive,
dQ = dQx + dQi ,
= dUx + dU, + PxdVx + p2dV, = 0 . (12)
Finite quasi-statical adiabatic changes are simply continuous se-
quences of equilibrium states and therefore are curves in the phase-
space (i.e., the />, V plane for a single body) which satisfy at each
point equations of the form (ii) or (12). Equations (ii) and (12)
are called the "equations of the adiabatics.”
If we consider i7 as a function of V and t, then
Hence (ii) takes the form
“ ( 1 ^ H- ^) dt = o. (14)
Equation (12) has interest only when the two bodies are in. thermal
contact. The system then can be described by three independent
variables, V^, V^, and t, the common empirical temperature:
(is)
Kp^, E.) = KP., F.) =;.
THE LAWS OF THERMODYNAMICS
17
Equation (12) can then be written as
dV^ +
(16)
Equations (14) and (16) are the equations of the adiabatics. Equa-
tions of the form (14) and (16) are called “Pfaflian differential equa-
tions."” We must now study some mathematical properties of these
differential equations.
8. Mathematical theorems on Pfaffian dijferential equations . — ^We
shall consider first a Pfaffian differential expression in two variables
.r and y:
dQ = A(.v, y)dx + ]'(.v, y)dy , (17)
which has the same form as equation (14). The integral of dQ be-
tween two points I and 2 depends in general on the path of the
integration. Hence I dQ
r«
cannot in general be written as y,)
““ y,), which means that dQ is not ‘ffntegrable.” This in turn
means that dQ in general is not a ])erfect differential of the function
y). If dQ were a perfect differential, we should have dQ = dc,
when* a is a function of ,v and y; we should have further
d(T
()cr
Ox
dx +
(ItT
i)y
dy .
('omi)aring (17) and ( iS), wi* havi*
or
-V(.v, y) =
On
Ox
v)
Off
Oy ’
/),V _ O'ff _ ('M'
<')y ff.vOy ().v
(18)
(>9)
(20)
('oiiilition (20) l)clwt‘cii tlic cocIl'K'icnls in lli(‘ Pfallian t‘X])rc‘Hsi<)n
need not, of coiirsi', he true.
i8
STUDY OF STELLAR STRUCTURE
Corresponding to (17), the Pfaffian equation in two variables is
or
dQ = Xdx -\-Ydy = 0,
(21)
dy ^ _X
dx V
(22)
The right-hand side of equation (22) is a known function of x and y,
and hence the Pfaffian equation (21) defines a definite direction at
each point in the (x, y) plane. The solving of the equation simply
consists of drawing a system of curves in the (x, y) plane such that
at any point the tangent to the curve (at that point) has the same
direction as that specified by (21). Hence, the solution of the equa-
tion (21) defines a one-parametric family of curves in the {x, y)
plane. The solution can therefore be written as a(x, y) = c = con-
stant. Then
da da dy
dx dy dx
o .
(23)
From (22) and (23) we easily find, that
„ 0cr _ ^ dcr __ XY
dx dy T
(24)
where t{x, y) is a factor depending on x and y. Equation (24) can
also be written as
X
(25)
Inserting (25) into (17), we have
or
i.e., if we divide the Pfaffian expression (17) by t, we obtain a
perfect differential. A factor, r, wfhich has this property is called
THE LAWS OF THERMODYNAMICS 19
an integrating denominator.” A Pfafl&an differential expression,
then , in two variables always admits of an integrating denominator.
If we replace a by another function of o-, say 5 [(r(jt:, y)], then S =
0 — constant will again represent the solutions of the differential
equation. In that case
, _^dQ
d<J dc T
(28)
= tdry) > (=“ 9 )
where
T(x, y) = y) . (30)
'rherefore, T is also an integrating denominator. Hence, if a PfaflBan
exjiression admits of one integrating denominator, it must admit of
an infinity of them. This result is easily seen to be true for a Pfaffian
expression in any number of variables.
We shall now proceed to consider a Pfaffian expression in three
variables. ('Fhc generalization to more than three variables is im-
mediate.) Consider the Pfaflian expression
dQ = Xdx + Ydy -h Zdz , (31)
where .V, and Z arc functions of the variables y, and z. Our
tluTmodynanncal equation (16) is of this form. The ratio dx:dy\dz
(lellnes a delinite direction in the {x, y, z) space. The equation
d{) = o, corresponding to (31), specifies that dx, dy, and dz must
satisfy a linear equation at each point in the space, and hence
s])(‘cifies a certain tangential plane at each point in the (a?, y, z)
space. A solution of a Pfaflian equation, dQ = o, passing through
a given point, (.v, y, z), must lie in the tangential plane through that
point; l)ut its direction in the tangential plane is arbitrary.
Now, (l{) in general will not be a perfect differential. If it were,
(/(j == da, whert‘ a is some function of x, y, z, so that
. , . d<r , , 5(r , , dc ,
,l{) = d<r{.x, y, z) = g^dx + + 9^ dz .
20
STUDY OF STELLAR STRUCTURE
Hence, by comparison with
Y
^~¥ x '
(31),
Y =
da _
dy ’
Z = —
^ dz ’
(32)
or
az _
dX .
dX_aY
(33)
dz dy ’
dx
dz ’
dy dx
The relations (33) need not be valid for arbitrary functions X, Y, Z.
But w,e can ask: Does the Pfaffian expression admit of an inte-
grating denominator? In other words, can we determine a function,
T, of X, y, and z such that
dQ
t{x, y, 2 )
j da , , flo’ j , da ,
^da = -^Jx+-dy^-dz
(34)
If we can determine an integrating denominator t{x, y, z), then
every solution of the differential equation dQ = o would also be
a solution of do- = o; or the solution can be written in the form
o'(*> y, s) = c — constant; i.e., the solutions can be any arbitrary
curve lying on any one of the one-parametric family of surfaces
cr(x, y, z) = c. It is, however, important to realize that we cannot,
in general, find integrating denominators for Pfaffian expressions in
more than two variables. This can be verified by the following ex-
ample. Consider the equation
dQ = —ydx -h xdy + kdz = o , (35)
where A is a constant. If the Pfaffian expression (35) admitted of
an integrating denominator r, then
-^dx + -dy + ^dz = da (36)
is a perfect differential. Hence, we should have
dc ^ X da ^ k
dx r ’ dy T ^ dz t '
53'\ T/ T 7^ dy dx\T)^T T^ dx'
(.^ 8 )
We have
THE LAWS OF THERMODYNAMICS
21
or
Bt , dr
2T = y IT •
dx ^ dy
(39)
Again
-(
az\
y\ ^ y ^ ^
t) dz doc \T J
k Bt
. (40)
or
dr y dr
Bx k dz ’
(41)
Similarly,
by\
'k\ kdr_ d /x\ _
^t) by dz\r/
X dr
T* dz ^
(42)
or
0T X Bt
By k Bz*
(43)
From (39), (41), and (43) we have r =0, thus leading to a con-
tradiction.
Ry means of such examples we realize that Pfaffian expressions
in three (or more) variables will not in general admit of integrating
denominators except under very special circumstances. It is neces-
sary to appreciate this, for precisely such special circumstances ob-
tain in thermodynamics.
We have seen that the Pfafhan differential expressions fall into
two classes, those which admit of integrating denominators and
those which do not. Wc must look for a less abstract characteristic
of this dilTcrcncc. Consider a Pfaffian equation in two variables.
'I'hen through every point in the {x, y) plane there passes just one
curve of the family aix, y) = c. Hence from any given point in the
plane wc cannot certainly reach all the neighboring points by means
of curves which satisfy the Pfaffian equation. We shall refer to this
circumstance by the statement that not all the neighboring points
arc accessible from a given point.
Now consider a Pfaffian expression in three variables. If it ad-
mits of an integrating denominator, the situation is the same as in
22
STUDY OF STELLAR STRUCTURE
tlie plane; aU the solutions lie on one or other of the family of sur-
faces <r{x, y, z) = c, so that we cannot reach all points in the neigh-
borhood of a given point. Only those points will be accessible which
are on the surface belonging to the family (r(x, y, z) = c, which
passes through the point under consideration.
We now ask the converse question : If in the neighborhood of a
point (however near) there are points which are inaccessible to it
along curves which are solutions of the Pfaffian equation, then does
the Pfaffian expression admit of an integrating denominator? Cara-
th6odory has shown that the answer to the foregoing question is in
the affirmative. The proof is as follows:
All those points which are accessible to a given point, Po (ac-
cessible along curves which are solutions of the Pfaffian equation),
and which are in its immediate neighborhood, must form, together
with Po, a continuous domain of points; hence we have three possi-
bilities : all the accessible points in the immediate neighborhood of
Po either fill a certain volume element containing Po, or a surface
element containing Po, or a line element passing through Po. The
first possibility is excluded because all points in a sufficiently close
neighborhood of Po would then be accessible to Po; this contradicts
our hypothesis that in the neighborhood of a point, however near,
there are always points inaccessible to it. Again, the last possibility
is also excluded because dQ = o = Xix -\-Ydy-\- Zdz already de-
fines an infinitesimal surface element containing only points ac-
cessible to Po. Hence, the points which are accessible to Po and
which are in its neighborhood must form a surface element, dFa.
If we now consider the boundary points P' of dPo, we can again
define surface elements dF' containing all the points accessible to
the points P' on the boundary of dF^. These surface elements dF'
must overlap dPo; at the same time the elements dF' cannot form
surface elements lying above or below dPo, for then along paths
going from Po to a point P' on the boundary of dFa, and thence
from P' along a curve lying in an appropriate element dF', we should
be able to reach all the points in an immediate spatial neighborhood
of Po; this would again contradict our hypothesis. Thus, the ele-
ment dP„, together with the elements dF', must form a continuous
set of surface elements. By this process of continuation, only points
THE LAWS OF THERMODYNAMICS
23
lying on a definite surface passing through Po are obtained, and
hence all the points accessible to Po must lie on a definite surface P©-
If we now start at a point Pi not on Po, we must obtain in the same
way another surface Pi which cannot either intersect or touch the
surface Po. In this way we can construct a whole family of noninter-
secting surfaces Po, Pi, Pi, , continuously filling the whole
{x, y, z) space, such that only points on any given surface are ac-
cessible to points on the surface itself. These surfaces then form a
one-parametric family of surfaces, (r(x, y, s) = constant, such that
d<r = o implies dQ = o. Hence, we must have
dQ = r(x, y, z)da(x, y, z) ,
where
^ X ^ V ^ z
^ d(T c)a dcr ’
dx dy dz
We have thus proved Caratheodory’s theorem
// a Pfajfuin cxf^rcsshn
d{) = Xdx + Vdy + Zdz
(44)
(45)
has the f)rof)crty I ha I in every arbilrarily close nciy^hhorhood oj a point P
ihcrc arc inaccessible points, i.e., points udiich cannol he connected
to P along curves which satisfy the equation (l(^) = o, then the Pfqfflan
expression miisl admit of an integrating denominator.
It is easily seen that the foregoing theorem must also })e true for
Pfalfian ex])ressions in more than three varialdes. t’urlher, it is clear
that, if a Ffallian ex[)ressi()n admits of one integrating denominator,
it must admit of infinitely many integrating denominators.
Vov the family of surfaces, o-(.r, y, z) == constant can also be writ-
ten as Slaix, y, z) =] constant, where S(o-) is an arbitrary func-
tion in <r. 'rhen we have
dS
dS dS d{)
da ~ da T '
(46)
or
dq = T{x, y, z)dS ,
(47)
24
STUDY OF STELLAR STRUCTURE
where
^ dS ^ ^ ^
dx dy dz
(48)
• Caratheodory ’s theorem, which expresses the mathematical equiv-
alence of the inaccessibility along curves dQ = o with the existence
of an integrating denominator t(x, y, z) to Q, contains, as we shall
see, the essence of the Second Law of Thermodynamics.
9. The Second Law of Thermodynamics. — The physical basis for
the second law is the realization that certain processes are not physi-
cally realizable. The most sweeping statement of this character is
that without ^‘compensation” it is not possible to transfer heat from
a colder to a hotter body; more precisely, the law is included in
Kelvin’s principle, which states: In a cycle of processes it is impos-
sible to transfer heat from a heat reservoir and convert it all into ivork^
without at the same time transferring a certain amount of heat from a
hotter to a colder body. The second law is sometimes also stated in
the form: It is impossible that, at the end of a cycle of changes, heat
has been transferred from a colder to a hotter body without at the same
time converting a certain amount of work into heat. "J'his latter state-
ment of the second law is due to Clausius. However, the essential
point of Caratheodory ’s theory is that it formulates the facts of ex-
perience in a very much more general way, enabling us at the same
time to obtain all the mathematical consequences of the second law
without any further physical discussion. In fact, in order to obtain
the full mathematical content of the second law, it is sufficient that
there exist certain processes that are not physically realizable. Cara-
theodory states his principle in the following form: Arbitrarily near
to any given state there exist states which cannot be reached from an ini-
tial state by means of adiabatic processes.
From Caratheodory ’s principle it follows in particular that there
exist states neighboring a given one which cannot be reached by
means of quasi-static adiabatic processes.
In the first instance we shall only apply Caratheodory ’s principle
to quasi-static adiabatic processes. Later (§ 10), we shall have oc-
casion to use the principle in its wider form, namely, that there exist
THE LAWS OF THERMODYNAMICS
2S
states neighboring a given one which are inaccessible to it along
nonstatic adiabatic processes.
From the restricted form of Caratheodory’s principle, it follows
that there are states neighboring a given one which cannot be
reached along adiabatics (Eqs. [14] and [16]); hence, by Carath6o-
dory’s theorem the Pfaffi'an differential expression for dQ must ad-
mit of an integrating denominator:
dQ = Tdc . (49)
For one single substance whose state is characterized by the two
parameters V and /, Caratheodory’s principle docs not lead to any-
thing new, because a Pfaflian expression in two variables always
admits of an integrating denominator.
When, however, we consider a system composed of two bodies
adiabatically inclosed and in thermal contact, Caratheodory’s prin-
ciple asserts something new in so far as we can now assert that
dQ = (/(),+ can always be written in the form
dQ = dQ, + dQ, = V„ l)da{V„ 1 ) .
(so)
On the other hand, we have for each of the two bodies
(IQi = li)dat{Vt, /,) ,
(SI)
dQi = Ti{ViJ.^d(r,{Vi, /i) .
(S2)
If the two bodies are in thermal contact, we have
/, = /. = /.
(S3)
Hence,
rda = 7,da, + Tida, .
(S4)
If we now choose a,, and / as the independent variables, instead
of F,, Vi, and /, we can regard t and a as functions of cr,, and t;
from (54) we then have
26
STUDY OF STELLAR STRUCTURE
From the last equation it follows that <t is independent of t; hence,
<T depends only on Ci and o-,, or
<7 = <r((ri, o-j) . (56)
From the first two equations in (55) it follow^ that Ti/t and t^/t
are also functions independent of t. Hence,
or
(57)
I dji _ _ I ^
Ti dt Tj dt T dt ’
(S8)
Now T, is a function only of <r, and t, and is a function only of
(T, and /. Hence, the first equality in (58) can be valid only if the
two quantities are functions of t only. We can therefore write (58) as
d log Ti a log T, a log T
ar - - ar = 'dt " “ ’
(S 9 )
where g{t) must be a universal function, because it has the same
value for two arbitrary systems and also for the “combined” sys-
tem. We are thus led to a universal function of the empirical tem-
perature, t.
From (59) we have, on integration,
log T = ^g{t)dt + log S(tr„ <r,) , (60)
log Ti = fgiOdt + log 2i((Ti) , (*=1,2), (61)
where the constants of integration S and Sj are independent of t and
are functions only of the other physical variables characterizing the
system. Equations (60) and (61) can also be written as
T = 2(<ri, <ri) . ^
Thus, for any thermodynamical system the integrating denominator
consists of two factors, one factor which depends on the tempera-
THE LAWS OF THERMODYNAMICS
27
ture (and which is the same for all substances) and another factor
which depends on the remaining variables characterizing the system.
We therefore introduce the absolute temperature, T, defined by
r = cc^‘'>'", (63)
where C is an arbitrary constant (instead of which we can also in-
troduce an arbitrary lower limit to the integral in the exponent in
[63]), and which is determined in such a way that two fixed points
(e.g., the freezing- and the boiling-point of water) differ by 100 on
the absolute scale. It should be noticed that T does not contain any
additive constant -in other words, the zero of the absolute scale of
temperature is physically determined. From (40), (62), and (63) we
have
dQ = rda — T da , dQi = ridai = T dai . (64)
If we are dealing with a single homogeneous body the state of which
is defined by the independent variables i and cr,, then depends
only on so that we can introduce the function 5 ,, which is de-
fined as
S, = i:,((r,)</(T, + constant . (65)
'I'he function 5 , depends only on <r, and is determined apart from
an arbitrary additive constant. Furthermore, St is constant along
an adiabatic. "J'he function .S',, so defined, is called the “entropy.’^
One can now write
dQt = TdSt . ( 66 )
If we now consider a system composed of two bodies in thermal
contact, we have for the two bodies separately
dQi = Tidot = i
r </a. = yvw. ,
(67)
dQi = TjJffa = i
r (la, = TdS, ,
(68)
28
STUDY OF STELLAR STRUCTURE
and for the combined system
dQ = rdff = J d<r{a,, <r,) , (69)
= dQ. + dQ» = + (69')
Hence,
S(<ri, ff^da — "^lia^dth + Sa(<r2)do’2 . (70)
From (70) it follows that
S(a'i, (Ti) = Si((ri) ; ^(o'l) o'a) = Sj(ffa) . (71)
Hence,
_ as 3(r 3V _ . ^
5 ffa flffj dtTi ddiBdi °
^2 _ ^ a(r a^g . ^
ddi dffi dda ddiddi ° ’ yl^
From (72) and (73) it follows that the functional determinant
as ^ ^ ^ ^ a(s,_cr)
a^i ddz da, ddi a(g,, ffj)
is zero, and consequently S(gi, gj) contains the variables g, and g^
only in the combination g(gi, g2). We can therefore write
2 (o-i, (Ta) = S(( 7 ) .
(75)
Equation (69)
can be written as
dQ = rJ <7 = TdS ,
(76)
where
dg ,
(77)
or
•S' = ^^S(<7)Jcr + constant ,
(78)
THE LAWS OF THERMODYNAMICS
29
where 5 is now the ^Total” entropy of the system. From (67), (68),
and (76) we further have that
dS = dS, + dS, = d(S, + S2) , (79)
or, in words: the change of entropy of a system composed of two
bodies in thermal contact, during a quasi-statical process, is the
sum of the entropy changes in the two bodies separately.
By a suitable choice of the additive constant entering into our
definition of entropy wc can arrange so that
S ^ S^ + S,, (80)
or: the entropy of a system is the sum of the entropies of its differ-
ent parts.
Equation (76) contains the mathematical statement of the Sec-
ond Law of 'rhermodynamics, which follows as a purely mathemati-
cal consequence of the Carathcodory principle: The diferential of
the heat, dQ, for an infinitesimal quasi-slatical change^ when divided
by the absolute temperature 'F, is a perfect differential, dS, of the en-
tropy funclion.
'fhe essential differences between (47) and (76) should be noted.
In (47) T and .S' (and r and a) are functions of all the physical vari-
ables; while in (76), r and T depend only on the empirical tempera-
ture, /, which is the same for the different parts of the system; fur-
thermore, (T and .S depend only on the variables (cr, and cr,) which do
not alter their values for adiabatic changes; finally, T is a universal
function of /, and S is a function only of o-(or,, cr.).
We shall now show that the gas-thermometer scale, pV = /, de-
liiu's a tem|H*ralure scale j)roportional to the absolute temi)erature.
It should be emj)hasized that the usual assumption that pV = t
defines, apart from a constant factor, the absolute temperature scale
is logically unsound. To assume beforehand that the absolute tem-
perature scale should be precisely pV = I and not any other mono-
tonic function, / = f(pV), is to beg the question. We shall see that
we cannot identify pV « 7 ' without an appeal to the Second Law
of 1 'hermodynamics. To do this logically, we need to know the in-
ternal energy, U, as a function of the state of the gas. The experi-
30
STUDY OF STELLAR STRUCTURE
mental basis is the idealized Joule-Kelvin experiment, which shows
that, when a gas expands adiabatically without doing any external
work, the product pV (i.e., the gas temperature, t = J\pV]) does not
change. (It should be noticed that an appeal is made here to an ir-
reversible process. As Carath6odory has pointed out, it is necessary
at some stage to appeal to an irreversible process to fix the zero-
point of the absolute temperature scale.) It follows, then, from the
Joule-Kelvin experiment that U is independent of t. Hence, we can
write
U = U{t) ; pV= F{t) , (8i)
where t is the empirical temperature. For the differential of the heat
for a quasi-statical change, we have
dQ = dU-{- pdV = Fit) y-
Define a quantity, x, by the equation
log X = dl + constant . (83)
Equation (82) can be re-written as
dQ = Fit)d log xV (84)
Hence, we can choose Fit) as the integrating denominator
T = FQ) ; <7 = log xF . (85)
Equation (84) now takes the standard form
dQ = rdtr . (86)
We can, of course, choose the integrating factor in many other
ways. If
V* = <r*(<r) ;
(8S')
THE LAWS OF THERMODYNAMICS
31
equation (86) can be written as
dQ = r*d(r* . (86')
Hence, there is no a priori reason to choose t = F{t) = pV as the
integrating denominator. But we have shown that
(87)
is a universal function which is the same in whatever way we may
choose to define the integrating denominator. g{t), defined by (87),
is invariant to the transformations (85'). From our definition of the
absolute temperature (Eq. [63]) we have
T = = C/'XO = Cpv . (88)
'I'hus the absolute temperature scale agrees with the temperature
on the gas-thermometer scale.
b>om dQ = 7 W 5 , we find that
,IS = d log xK ,
(89)
or
S = log xF + constant .
(90)
If we write U = CyT and consider Cy as a constant, and further de-
line R = i/(‘, we have
log X = ^ ^ ^ constant .
(90
Hence, linally,
.S = .S;, + r, log 'f + 7 ^ log V ,
(92)
where .S',, is a constant.
10. The l)rinci[?lc of the increase of cniropy. So far
we have con-
sidered only quasi-statical changes of state, though at one point
32
STUDY OF STELLAR STRUCTURE
(§ 9 ) we had to consider a nonstatical process when we appealed to
an idealized Joule-Kelvin experiment. We shall now discuss non-
statical processes more generally.
We shall consider, as we have done so far, an adiabatically in-
closed system composed of two bodies in thermal contact. The equi-
librium state of such a system can be characterized by three inde-
pendent variables, such as Fi, F 2 , ^ (the variables we have used so
far). We shall now choose Fi, F^, and S as the independent vari-
ables. Let F;, VI and 5° be the values of the physical variables in
an initial state and Fi, F 2 , and 5 in a final state. We now assert
that S is either always greater than 5° or always less than S°.
To show this, we consider the final state as being reached in two
steps:
a) We alter the volumes FJ and VI by means of a quasi-statical
and adiabatical process such that the volumes at the end are F i
and F 2 . In this way we keep the entropy constant and equal to 5".
b) We then alter the state of the system, keeping the volumes
fixed, but change the entropy by means of adiabatical but nonstati-
cal processes (such as stirring, rubbing, etc., in which dQ = o but
dQ 5 ^ TdS) such that the entropy changes from 5" to 5.
If, now, S were greater than 5® in some processes and less than
5 " in others, then it should be possible to reach every close neigh-
boring state, (F„ Fa, 5), of the initial state, (F", V], 5"), by means
of adiabatic processes. (After reaching the state (Fi, Fa, S), we can
reach all the states, (Fi, F', S), by means of processes [a]). This
contradicts Caratheodory’s principle in its more general form, which
postulates that in any arbitrarily near neighborhood of a state,.
(F;, FJ, S)j there exist adiabatically inaccessible states even when
we allow nonstatical processes. Consequently, by means of the proc-
esses ( 6 ), and therefore also by means of the processes (a) and ( 6 ),
the entropy S" of the system can either only increase or only de-
crease. Since this is true for every initial state, we see that, because
of the continuity of the impossibility of “increase” or “decrease,”
the entropy of the system we have considered must either never in-
crease or never decrease. The same must also be true for two inde-
pendent systems because of the additive nature of entropy. We have
thus proved : For all the possible changes {quasi-statical or otherwise)
THE LAWS OF THERMODYNAMICS
33
that an adiabatically inclosed system can undergo, the entropy, S, must
either never increase or never decrease.
Whether the entropy decreases or increases depends in the first
instance on the sign of C introduced in our definition of entropy (78).
This is naturally chosen in such a way that the absolute temperature
is positive. Then one single experiment is sufficient to determine
the sign of the entropy change. By the expansion of an ideal gas, G,
into a vacuum, the entropy Sq of the gas increases, as can be seen
from equation (92) {V increases and T remains the same). We now
consider a system composed of the gas, G, and of another body, K.
If we consider such changes of state in which the entropy Sk of the
body remains constant and Sq changes, then S = Sa + Sk must
increase (since, as we have just seen. Sc; always increases); conse-
quently, S can never decrease. Hence, if we consider processes in
which the entropy of the gas remains constant, it is clear that, as S
can only increase, Sk can only increase; this is true also when K
and G are adiabatically separated. Hence, in general we have proved
the following important result;
For an adiahalically inclosed system the entropy can never decrease:
S > S'' , (nonstatical ])rocess) , \ . .
S = S" (statical j)rocess) . I
It follows that if in any change of slate of an adiabatically in-
closed system the entropy becomes <li(Terent, then no adiabatic
change can be realized which will change the system from the final
to the initial state. In this sense, therefore, every change of state
in which the entropy changes must I)e irreversible. This can also
be stated as follows: For an adiabatically inclosed system the en-
tropy must tend to a maximum.
Still another formulation of the foregoing is
^'ij! < o , (04)
where the integral is taken over a closed cycle of changes, it being
assumed that during the cycle the system can be characterized at
34
STUDY OF STELLAR STRUCTURE
each, instant by a unique value for T. To prove this let us consider
a cycle of changes in which the working substance is carried through
states A and B, and in which, further, the part of the cycle from
A to 5 is carried out adiabatically (but not necessarily statically)
while the part of the cycle from 5 to A is carried out reversibly. For
this cycle of changes
Since the part of the cycle from AtoB has been carried out adiabati-
cally, we have
ff-s:
= Sa — Sb :
which, according to (93), must be zero or negative. We have thus
proved (94) for the special cycle of changes considered. The argu-
ments can be extended to prove (94) quite generally.
We thus see that the full mathematical content of the second law
can be deduced from Carath6odory’s principle. But the question
still remains whether Caratheodory’s principle can lead us to Kel-
vin’s formulation of the second law. To answer this, we must supple-
ment Caratheodory’s principle with some additional axioms before
we can derive Kelvin’s or Clausius’ formulation of the second law.
The arguments necessary to establish this involve some rather deli-
cate considerations, and these go beyond the scope of our present
chapter. The interested reader may refer to an illuminating discus-
sion by T. Ehrenfest Afanassjewa quoted in the bibliographical note
at the end of the chapter.
II. The free energy and the thermodynamical potential . — ^We have
shown in § 10 that
pf ^ O . ( 95 )
where the integral is taken over a closed cycle of changes. Let us
suppose that the closed cycle of changes carries the working sub-
THE LAWS OF THERMODYNAMICS
35
stance through states A and B, and that, further, the part of the
cycle for 5 to ^ is along a reversible path. Then
or, since the path from J5 to A is reversible, we have, according to
( 95 ) and ( 96 ),
^ Sii - Sa . ( 97 )
Equation (97) is, of course, equivalent to (95).
Let us now consider an isothermal change. I'hen (97) can be writ-
ten as
^ T{Sn - Ba ), (98)
where 7 ' denotes the constant temperature. By the First Law of
'I'hermodynamics we now have.
Ub-Ua + Wab^ nSn - Sa) , ( 99 )
where Wab is the work done by the system. Equation (99) can be
written alternatively in the form
Fb - Fa + IV a B ^ o ,
where
F ^ U - TS .
(too)
(101)
The function F, thus introduced, is called the “free energy” of the
system. From (100) it follows that for an isothermal change in which
no work is done the free energy cannot increase.
Another function of imi)ortance is the thermodynamical i)otential,
defined by
Cl = pv — u pv y iS . (102)
It is clear that if the temperature and the external forces arc kept
constant (/ cannot increase.
36
STUDY OF STELLAR STRUCTURE
12. Some thermodynamical formulae . — So far we have concerned
ourselves only with general principles. We shall conclude this chap-
ter with the derivation of some thermodynamical formulae which
are of considerable practical importance.
Let us consider a homogeneous isotropic medium. Then for a
quasi-statical change (in Eq. [14] we shall now use the absolute
temperature, T, instead of the empirical temperature, t)
Since dQ/T is a perfect differential, we should have
6 f:
c/au , NT
a 1
^ au\
arLi
rVau"'"
aF'
dT)
or, carrying out the differentiations.
(104)
_ 4. j 4. 1 - ^ \
Vaf/Kj TavdT’
or
(106)
Let us next consider the free energy. By definition (Eq. [loi])
dF = dU - TdS - SdT , (107)
or, since
dQ = TdS = dU + pdV , (108)
we have
dF SdT - pdV . (109)
dF is, however, a perfect differential. Hence, we should have
(no)
THE LAWS OF THERMODYNAMICS 37
Finally, let us consider the thermodynamical potential, G. We
have
dG= dF+ pdV + Vdp ,
(III)
or, using (109),
dG = -SdT + Vdp .
(112)
Hence, we should have
(113)
We shall have occasion later to use (106), (no), and (113).
BIBLIOGRAPHICAL NOTES
C. Caratiieodory^s paper appeared in Maih, Ann.^ 67, 355, 1909. Further
developments are contained in C. Caratheudory, Siiz. Ber. d, Pril. Aktid.,
p. 39, Berlin, 1925; T. Ehrenfest Afanassjewa, Zs.f. Phys., 33, 933, 1925.
General expositions have been given by M. Born, Pliys. Zs., 22, 218, 249,
282, 1921, and by A. Lande, Ilandh. d. Pliys., 9, chap, iv, Berlin, 1926. Our
account is based largely on Born’s and Lande’s expositions.
CHAPTER II
PHYSICAL PRINCIPLES
In this chapter we shall be concerned with some misceUaneoiis
problems which form the background to the study of stellar struc-
ture. The main topics for consideration are: the thermodynamics
of a perfect gas, uniform expansion (or contraction) of gaseous con-
figurations, the virial theorem, and the thermodynamics of black-
body radiation.
I. The specific heats of a perfect gas . — We shall consider a perfect
gas for which, according to the results of the last chapter,
pV = RT-, U=V{T), (i)
where T is the absolute temperature and where the constant of pro-
portionality, R, so introduced, is called the “gas constant.”
For an infinitesimal quasi-statical change of state we have
dQ = dU + pdV , (2)
or, according to (i),
dQ = ‘^iTdT+pdV. ( 3 )
Let a be a function of the physical variables. Then the specific heat,
Ca, at constant a is defined by
Ca
a = constjint
(4)
The right-hand side of (4) is to be determined from (3) in such a
way that a remains constant. Thus, the specific heat, cy, at con-
stant volume is given by
Cv
dU
dT ■
(S)
To determine the specific heat, Cp, at constant pressure, we pro-
ceed as follows: From the equation of state we have
pdV + Vdp ■= RdT .
38
( 6 )
PHYSICAL PRINCIPLES
39
From (3) and (6) we have
dQ^{jf + ^dT - Vdp , ( 7 )
from which it follows that
= ( 8 )
Combining (5) and (8), we have the important result
Cj, — cv — R • ( 9 )
The ratio of the specific heats, denoted by 7, is defined as Cp/cy-
In further work, we shall assume that cy is independent of T.
This is a consequence of the kinetic theory of gases, to which we
shall return in chapter x. From (5) we have, then, for the internal
energy U:
U = evT . (10)
2. Adiabatic changes,- Using (5), we can write for the differen-
tial dQ for a quasi-statical change
dQ - cvdT + pdV , (11)
or, using the equation of the state,
(/() = c,7/7' + y rfK. ( 12 )
For a quasi-statical adiabatic change, therefore,
cvdr + '"^,iv = o, (i.O
or, using (g),
dT .dV , .
cy y, + ( 0 < — ‘•'r) y = o , ( 14 )
from which wc obtain
(is)
cy log 7" + (cj) — cy) log V = constant .
40 STUDY OF STELLAR STRUCTURE
In terms of the ratio of the specific heats we can re-write (15) as
j'77-i = constant .
(16)
Using pV = RT, we can eliminate T in (16) and obtain
= constant . (17)
Similarly, by eliminating V between (16) and (17), we have
pi-y Ty = constant . (18)
Hence, along an adiabatic we have
pyy = constant ; Ty = constant ; TVy~' = constant . (19)
The foregoing equations (19) are due to Poisson. The derivation in
the form given above is due to Lord Kelvin.
3. Polytropic changes . — A polytropic change is a quasi-statical
change of state carried out in such a way that the specific heat re-
mains constant (at some prescribed value) during the entire process.
Thus,
dQ
if
c = constant .
(20)
An adiabatic, then, is a poly tropic of zero specific heat, and an iso-
thermal a poly tropic of infinite heat capacity. It is also clear that
quasi-statical changes in which the pressure and the volume arc kept
constant are polytropics of specific heats Cp and cy, respectively.
Poly tropic changes were first introduced in thermodynamics by
G. Zeuner and have been used extensively by Helmholtz and cs- j
pecially by Emden.
From (ii) and (20) we have, for an infinitesimal polytropic
change,
{fiv — c)dT pdV = o . (21)
Equation (21) is the equation of a polytropic. From (21) we have
PHYSICAL PRINCIPLES
41
or, integrating, we have
= constant .
(23)
We shall define the polytropic exponent, 7', by
f Cp c
7 = - .
Cv — c
(24)
We have
-y' _ I = .
(25)
cy — r
Equation (23) can then be written as
T'l/'r'-t = constant ,
(26)
which is of the same form as (16) except that the poly tropic expo-
nent, 7', replaces the ratio of the specific heats, 7. Hence, quite
similarly, as in the last section, we have that along a poly tropic
pV*' = constant ; p^-y ' = conslanl ; = constant . (27)
4. A theorem due to Emden,
“ Let AB and Cl) be two
polytropics of heat capacity
r, and exponent yr, further,
let AD and BC be two other t
poly tropics of heat capacity P
and ex])onent 7^. Let these
four i^olytropics intersect at
the j)oints A, B, C, and I),
Let Va, and be the
values of the j)hvsical vari-
al)les at A; />ii, V n, and
T/i, the values at B; and
hr,. A
so on.
Consider the case in which the gas goes through the cycle A BCD
(luasi-statically. Since d{)/'J' is a i)erfect diffiTential,
(28)
42
STUDY OF STELLAR STRUCTURE
over a closed cycle must be zero. We shall evaluate the foregoing
integral for the cycle under consideration. Since dQ = CidT along
AB and CD, and dQ = CadT along AD and BC,
rdQ rST , C^dT, rdT , ^ dT , ,
(30)
or
, Tb , , Tc , - ^ Ta
Cl log ^ + Ca log ^ + Cl log yr + C2 log ^ = o ,
or, again,
(c.-c.)log^- = o
Since c, ^ c„ we have
TbTd = T^cTc ,
(31)
(32)
or
Ta Td
Ta Tb
(33)
Tb Tc ’
Td Tc '
Since along the poly tropics AB
(27) with y' = Y„ we have
and CD we have
the relations
Ta ^
VT"
(34)
Tb
Similarly,
Td
F?-'
(35)
Tc
F 5 ‘"‘ ■
Combining (33), (34), and (35), we have
Va Vd
Va Vs
(36)
Vb Vi ’
Vd Vc •
Similarly, we can show that
pA _ pD ,
pB pc ’
II
(37)
Thus we have proved : If a pair of polytropics belonging to a given
class (i.e., a given exponent) is intersected by another polytropic belong-
PHYSICAL PRINCIPLES
43
ing to O’ dijfcr&nt doss, then the votio of the physical vofiobles (p, V,
or T) at the points of intersection are the same whatever polytropic be-
longing to the second class we may choose.
We can state the foregoing theorem, due to Emden, in the fol-
lowing somewhat different way : A polytropic AB of exponent y , is
cut at the point A by another polytropic AD belonging to another
class, of exponent y^ (Ya arbitrary but different from 7.)- Along
AD we consider the point D such that Pa! Pd (or Ta/Tb or VaIVd)
is some fixed constant. We now allow AD to be any polytropic be-
longing to class 7,. The locus of D is then another polytropic, be-
longing to class 7i. We shall sec that, stated in the foregoing form,
Emden’s theorem has an important application to the theory of
gaseous configurations.
S. Polvlropic temperature and the Emden variables. As we saw
in § 3, along polytropics belonging to the class y' we have
pV'>' = constant ; TV^'-' = constant ; />■->'' 7 ^' = constant . (38)
In the ip, V) plane, the polytropics belonging to a given exponent 7'
form a one-parametric family of curves, the parameter being the
“constant" occurring in the lirst of the foregoing formulae. 'I'his
family of curves can be classified by labeling each curve by what
Emden calls the appropriate “ixilytropic temperature”; the latter
is defined as the temperature along the given polytropic where the
specific volume, V (and therefore also the density), has the value
unity. We shall use Oy to denote the polytropic temperature. 'I’hen,
7 'l V-' = oy . I = ()y . (39)
Since the isothermal is a i)olylroi)ic of infinite heat capacity, 7' = ' 1
and we have
T = ()„,, (40)
i.e., the polytropic temperatures for the isothermals agree with ac-
tual temiieratures labeling the isothermals.
In terms of the poly tropic temiK'rature we can represent the
jihysical variables very conveniently. Let us write
p = Xfl" ; » = .y/ 1 j . ( 41 )
44
STUDY OF STELLAR STRUCTURE
where X is some constant factor to allow for a change in the scale on
which density is measured; n introduced as above is called the “poly-
tropic index.” Since the density, p, is the reciprocal of the specific
volume, F, we have, from (39), that
T = . ( 42 )
If we choose X to be unity, we see from (42) that 9 is the temperature
in a scale in which the polytropic temperature is unity. We further
have
p = RpT = ^
If we consider polytropics with zero specific heat, then we have
adiabatics as a special case. Then 7' = 7, and we have definitions
for the adiabatic temperature and adiabatic index.
6. Entropy changes . — ^We have
dQ dT , p
dS = = cy-^ +^dV ,
(44)
or, using the equation of state.
Co
II
1
1
( 45 )
Now
II
(46)
By differentiating (46),
¥ - fr' + (V - ■) ■
1 By' p
(47)
Inserting (47) in (45), we have
,S.cy[fi+W-y)'‘^\.
(48)
or
‘S' = So + C7[log -h (7' - 7) log p] . (4g)
For an adiabatic 7' = 7, and
S = So + log 0-y . (50)
PHYSICAL PRINCIPLES
4 S
Again, for (48) we have
dQ = TdS = + ( 7 ' - 7 ) , (sO
from which wc obtain
{dQU = cvT'^. (52)
Hence, the withdrawal of heat lowers the adiabatic temperature,
while the supply of heat increases the adiabatic temperature.
If wc consider changes along a given polytropic, then 0 y does
not change; and hence, by (47) and (48), along a polytropic,
y = (7'-I)'^p^ (S3)
(S 4 )
I'kiuation (54) could have been dcrivccl directly from the definition
dQ = cdT. Along an adiabatic dS is, of course, zero.
7. Uniform expansion and conlraclion of {gaseous configurations;
cosmogcnclic changes.- Consider a perfect gas configuration in gravi-
tational equilibrium. 'J'hen
, , ( 55 )'
dr r‘
where r denotes the radius vector with the center of the configuration
as origin, h'urthermore, M(r) is the mass inclosed inside a spherical
surface of radius r, and p = RpT. 'I’he foregoing equation is an ele-
mentary consequence of hydrostatic equilibrium. ('I'hc meaning of
I55I is further commented upon in chap, iii.) We shall refer to a
perfect gas configuration satisfying (55) as a “gas sphere.”
An expansion or contraction of a- spherical distribution of matter is
.said to be uniform if the dislaiuc between any two points is altered in
the same way as the radius of configuration.
' In wriliiiK lliis o(iiuitiiin wc have iici'lcctcil nulialii)n preswure (see chaps, iii and vi).
46
STJUDY OF STELLAR STRUCTURE
Let the radii* of the initial and the final configuration be Ro and
Ri, and, further, let
Ri = yRo .
(S6)
Then, if r, and To are the distances of any specified element of mat-
ter from the center before and after the expansion,’
fi = yr« (S 7 )
if the e 35 )ansion has been carried out uniformly. More generally, if
an element has an extension dso in the initial configuration, then it
will have an extension ydso after the uniform expansion:
dsi
In particular,
dri
yds „ .
(S8)
ydr , .
(S8')
Let po, po, To and p,, Ti he the density, pressure and tempera-
ture at “corresponding points” (i.e., at a distance u in the initial
configuration and at a distance r, = yro in configuration after ex-
pansion). It is clear that
Pt = y ’Po ,
(59)
since the corresponding volume elements in the two configurations
are in the ratio y’, while the mass inclosed in either is the same.
We shall now consider a uniform expansion of a gas sphere. Then,
we should have
dp^ _ _GM(r„)
dr„ ' rl ’
(60)
dpi _ GM(ri)
dr,~
(61)
Since, however, M(fo) = M(rt), we have, according to equations
(58'), (59). and (61),
dp.
^ GM(r„) _
yV2 ^
Po • ydr„ = pjr„ . (62)
At the boundary of the configuration, p, p, and T are all zero. The vanishing of
p, p, and T defines the radius of a gas sphere.
3 For brevity we shall explicitly refer only to “expansion” and not repeat each time
“expansion or contraction.”
PHYSICAL PRINCIPLES
47
By (6o), then,
dpi = y~‘^dp„ ,
from which it readily follows that
pL = y~*Po .
Since p = RpT, we have, from (59) and (64),
pi = RpiTi = Ry-^paTi = y~*Po = y~*RpoTo ,
or
Ti = y-'Ti, .
Equations (56), (59), (64), and (66) can be written as
p, _ V„ _ p, ^ (Ro\\ Ti ^ R„
p„ Vi \Ri) ’ P« \Ri) ' T'o Ri
(63)
(64)
(65)
( 66 )
(67)
We have thus proved the following theorem: By a uniform ex-
pansion (or fonlradion) of a sphere, the dcfisity, pressure, and tem-
perature at every point alter according to the inverse third, fourth, and
unit power, respectively, of the ratio of the initial to the final radius.
'I'he theorem in this general form is due to P. Rudzki (1Q02), though
in a less general form it was known to Homer Lane (1869) and also
to A. Ritter (1878). We shall refer to the foregoing theorem as
“Lane’s theorem.”
Since the heat energy is proportional to Cy 2 ', a further consequence
of Lane’s theorem is that the total heat energy in a gas sphere varies
inversely as the radius during the process of uniform expansion.
For an inlinitesimal uniform expansion we clearly have
dT _ ^ dp _ I dp _ _ dR„
T ,3 p 4 /» Ro '
h’rom (67) it follows that
P-
P«
( 68 )
(69)
TiV'/' = 7’„r
■ /.« .
II }
Alternalively,
piV'/' = A, FA''' ;
r,Pi . (70)
48
STUDY OF STELLAR STRUCTURE
Thiis, if a gas sphere expands {or contracts) uniformly through a se-
quence of equilibrium configurations, then the matter at every point un-
dergoes a polytropic change belonging to the exponent 7' = 4/3, or
n = 3. This result is due to Ritter, who was thus the first to rec-
ognize the special “cosmological” importance of poly tropic changes
of exponent 4/3. For this reason, he called polytropic changes of
index 3 “cosmogenetic changes.”
Since, according to Ritter’s theorem, the physical variables change
along a cosmogenetic during a uniform expansion, we can apply the
results of § 6 to calculate the corresponding change in entropy. The
appropriate formula to use is (54) with 7' = 4/3. Hence, for an in-
finitesimal expansion, the change in entropy, dS, is given by
dS = cK 4 - 37 )f’,
(71)
or, by (68),
dS=- cr (4 - 37) f-" .
XVo
(72)
Further, we have
dQ
= rds= -CKr( 4 - 37 )‘^".
(73)
8. Uniform expansion {or contraction) of polytropic gas spheres . —
If, in a gas sphere, the pressure and density are related according to
equations (41) and (43) with some definite value for Gy, then the
gas sphere is said to be a “polytropic gas sphere of index or
more simply as a “poly trope of index «.” This means that, if we
plot the pressures at the different points in the gas sphere against
the specific volumes at the respective points, then the points must
all lie along a definite polytropic of index n and exponent 7’. Let
us fix our attention on one definite point, A„, on this polytropic.
Through this point A„ draw a polytropic of index 3. By Ritter’s
theorem, the effect of a uniform expansion (or contraction) is to
displace the point A^ along the polytropic of index 3 to another
point, At, such that the temperature at A, bears a constant ratio
{Ro/Ri) to the temperature at A^. Now this happens to all the
points, A„, on the original polytropic as a result of the uniform ex-
pansion. By Emden’s theorem (§ 4), we obtain in this way another
PHYSICAL PRINCIPLES
49
polytropic of index n with a polytropic temperature different from
the polytropic temperature of the original gas sphere. We thus see
that the values of p and V at the points in the new configuration
again lie on a polytropic of index n. Thus, the configuration resulting
from the uniform expansion of a polytropic gas sphere is again another
polytropic gas sphere belonging to the same index.
As we have just seen, the polytropic temperature of a polytrope
is altered as a result of a uniform expansion. Let G,,' (o) and G^,- (i)
be the poly tropic temperature before and after the expansion. Then
by (42) and (67),
or
(V(i) ^ 7'.F7'~'
= ^-(4-37') ^
(74)
ov(o)/?r”' = (^Y'CO^r"' •
(7S)
Hlmicc, for an inlinilcsimal expansion,
n . = -(4 - 37 ) » . (76)
( )y* Ki)
I^Yom f 4 (S) we can now calculate the change in entropy for an inlini-
tesiinal ex])ansion of a ])()lytrope. Since
(IS =
dOy' , dp
i)y
+ (7' - 7)
we have, according to (j()) and (08),
dS = ~crl (4 - 37') + 3(7' - 7)1 = -^■r (4 “ 37 )
All
(77)
(7«)
thus recovering our earlier result ( 72 ).
9 . 77/c viridl Ificorem. We shall now consider the general motion
of a cloud of particles, d'he “particles” may be gaseous molecules,
dust |)articles, or even stars.
Let m denote the mass of a particle; .v, y, and z its co-ordinates;
and .V, L, and / the components of the force acting on it. 'Then,
by Newton’s laws of motion.
m
d\x
dl^
X
d'y
df - '
(I Z y
(70)
so STUDY OF STELLAR STRUCTURE
We have
or, using the first of the equations (79),
= + (81)
Similarly,
= »«(fy + >-F, (82)
(83)
Adding the foregoing equations, (81), (82), and (83), we obtain
i I (w) - »[( J)’ + (*)■ + (^')'] + (Adr + jy + «z) . (84)
The first term on the right-hand side is simply twice the kinetic
energy of the particle. Hence, summing the foregoing equation over
all the particles, we have
l2^=^T+^ixX + yY + zZ), (8s)
where I is the moment of inertia about the origin defined by
I = 2 (^ 7 *") , (86)
and T is the kinetic energy of motion of the particles forming the
cloud. The second term occurring on the right-hand side of (85) is
called the ‘Virial of Clausius.’’
To evaluate the virial, we fix our attention on two specific par-
ticles of masses Wi and m2 at the points y^, Zi) and (x2j yz, z-z).
Let the force exerted by the second on the first have components
A, B, and C, so that the force exerted by the first on the second will
have the components —A, —B, and — C. The contribution of this
pair of forces to the virial is given by
A{xi - X2) + Biyi — ya) + C(zi - Zz) ,
(87)
PHYSICAL PRINCIPLES
51
and hence
Virial = - x^) + B(yi — 3/2) + C(si— Za)] , (88)
where the summation is extended over all the pairs of particles. For
a cloud of density so low that the ideal gas laws may be assumed to
hold, all forces except the gravitational forces may be neglected.
Thus we may take for A, 5 , and C the components of the force
(where G is the constant of gravitation) directed from
2 to I. Hence, the components are
and
—
A*2
rxi
)
—
Tm
S|
—
r,2
»
along tire X-axis
along the F-axis -
along the Z-axis
Virial
(89)
(90)
Now each term inside the summation sign is simply the work done
in separating the pair of particles to infinity against the gravitational
attraction. Hius the virial is seen to be the potential energy, of
the cloud of particles under consideration. Hence, we have
1 (H
2
27 + 12 ,
(91)
an equation derived by Poincare and Eddington. If the system is
in a steady state, / is constant, and consequently we have
27 + 12 = o . (92)
Equation (02) exjrresses what is generally called the “virial theo-
rem.”
10. A n applicalion of Ihe virial Uieorcm. Let us apply the virial
theorem to a perfect gas con i'lgu ration in gravitational equilibrium.
Consider an element of mass dm at temi)erature T. I'Vom the kinetic
theory of gases (sec chap, x) the mean kinetic energy of a single
52
STUDY OF STELLAR STRUCTURE
molecule in this element is ‘where k is the Boltzmann constant.
Let there be dN molecules in the element of mass under considera-
tion. The contribution to the kinetic energy (of molecular motion)
due to this element of mass is given by
dT = ^kTdN = \RTdm = |(cp — cv)Tdm . (93)
But the internal energy, dU, of the element of mass is given by
(Eq. [10])
dU = cvTdm . (94)
Hence,
dT = |(t - T)dU , (95)
or, for the whole configuration,
T = ^irr-i)U. (96)
By the virial theorem, then,
3 (y - T)U + £2 = 0 . ( 97 )
Let E be the total energy. Then
17 + £2 = £ . . ( 98 )
From (97) and (98) we easily obtain
£= -( 37 - 4)17 = ^^^^£ 2 . ( 99 )
The foregoing equation has the following consequences:
a) For a mass of gas for which 7 = 4/3, we see that E — o in
a steady state (independent of the radius of the configuration). A
small radial expansion of the mass is, accordingly, possible, the mass
changing from one equilibrium configuration to an adjacent con-
figuration of equilibrium without change of energy. It follows that,
if we consider a sequence of equilibrium configurations in which 7
varies continuously, then at 7 = 4/3 a change from stability to in-
stability (for radial oscillations) must set in.^ On the other hand, we
4 This is intuitively obvious, but for a general discussion see J. H. Jeans, Problems
of Cosmogony and Stellar Dynamics, pp. 20-23, Cambridge, 1919.
PHYSICAL PRINCIPLES
S3
see that for 7 = i, S2 = o for any prescribed E; i.e., for 7 = i no
stable configuration is possible. Hence, it follows that we have
“stable” gas spheres only for 7 > 4/3.
This result is originally due to Ritter and Emden; our proof,
however, is due to Poincar6.
b) For 7 > 4/3, equation {99) shows that E must be negative;
or, in other words, in a steady state the energy is less than in a state
of diffusion at infinity. Suppose, now, that the configuration con-
tracts so that the potential energy changes by an amount Ml. If
AE and AU are corresponding changes in the total energy, E, and
the internal energy, U, then by (99)
AE= - (37 - 4) AU = Z i)
Hence, the amount of energy lost by radiation is - AE:
-AE= — f' ~ 4 : AQ , (loi)
3(7 - i)
which is positive for a contraction of the configuration. At the same
time, the internal energy increases by an amount
AU = 7-^ r Ail , (102)
3(7 - i)
which is again jiositive for a contraction. 'I’he reason for the increase
in the internal energy consequent to a contraction of the configura-
tion is that of the work j Aflj done by contraction, only the fraction
_ 4)/;5(7 - 1)] is lost in radiation to space outshle, and the
remaining fraction [i — (37 — 4 )/.^iy ~ 01 = ~ i)] is used
in raising the temperature of the mass.
II. The Slejan-Iiollzmami law. We shall now conshler the appli-
cation of thermodynamics to inclosures containing radiation. Con-
sider a perfectly black body. Af, contained in an inclosure with per-
fectly relied ing walls. 'I'he inclosure will be traversed in all direc-
tions by radiation. Let the temperature of the black body contained
in the inclosure be T. In a steady state, the inclosurc is traversed by
“black-body radiation” at temperature T. We shall assume that
54
STUDY OF STELLAR STRUCTURE
quasistatical processes can be carried out with the radiation. We shall
suppose, further, that the radiation is the same throughout the in-
closure. Let the energy of radiation per unit volume be u, so that
the internal energy JJ will be uV:
U = uV . (103)
There is a certain analogy between radiation and a perfect gas.
The energy of both depends on temperature, and both exert pressure.
According to the electromagnetic theory of light, radiation exerts
the pressure
p = \u . (104)5
Let us allow the inclosure to expand quasi-statically while the
temperature is maintained constant. Let the volume, V, increase
by an amount dV while u and p remain unaltered. Consequently,
the internal energy, U, increases by an amount tidV, so that
We shall now use the thermodynamical formula established in the
last chapter (Eq. [106], i) :
(106)
In our present case p depends only on T; and hence, according to
(104) and (105), we can write (106) as
1 7^ 1
« = -sZ
(107)
or
rp du
^ dT ’
(108)
or again,
u = aT^ \ p = \aT* .
(109)
s This is proved ia chapter v, which deals with radiation problems in greater detail.
Here we are only concerned with one straightforward application of thermodynamics
to inclosures containing radiation.
PHYSICAL PRINCIPLES
SS
Thus, the energy of black-body radiation per unit volume is pro-
portional to the fourth power of the temperature. This is the state-
ment of Stefan’s law. The constant a introduced in (109) is called
the “Stefan-Boltzmann constant.” (Stefan empirically discovered
the law in 1879 and Boltzmann gave the proof [essentially the one
given here] in 1884.) , . j-
12 Adiabatic changes in an inclosurc containing matter and radia-
tion.-{a) We shall first consider the case of an inclosure containing
radiation only. For a quasi-statical change,
dQ^ dU + pdV ,
(no)
or, by (io,0 (*o 4 )j
dQ =
d{nV) - 1 - ludY = •
(no')
For a quasi-statical adiabatic change, then,
vdit + = 0 »
(in)
or
i/4'.i/.i = constant ,
(112)
or, since u == (iT'\
yq'i/.! = constant .
(n,^)
I'roni ( io()) and ( i i
2) we have
= constant .
(114)
'I'hus radiation, in this resiiect, behaves like a perfect gas with a
ratio of the speeilic heats 7 — 4 /.b . . , v 1
b) Let us now consider an inelosure containing both matter and
radiation. We shall only consider the case where the matter is a
perfect gas. 'I’lie internal energy of such a system is, according to
(lo.O, (109), and (10),
// ^ .rl"/’! -I- rvT . (115)
Let p, denote the radiation iiressure and />„ the gas jiressure.
the total pressure, P, is, accordingly,
R
P = />. + /»» = + y ‘ ■
'rhen
(1 ifi)
S6 STUDY OF STELLAR STRUCTURE
For a quasi-statical change
+ + (.. 7 )
By (115) and (116)
(If )» - + cr- P.'j , (n8)
(ff)r ■
Inserting (118) and (119) in (117), we have
dQ = ril i2Pr ^ — p^dT + ( 4 /»r + Po)dy • (120)
i \ Cp Cv J
Let y be the ratio of the specific heats (t = Cpjcy). '.rhcn
dQ. = y^i2/>r + i;-- P^dT + {j^p, + pM^ ■ ('^i)
For an adiabatic change,
{^12pr + ^ + (4/’i- + Pii) y “ ° • ( >2-’)
We define the adiabatic exponents 1 ’,, Tj, and Tj by the relations
^2^
+
II
0
(12.0
dP 1% dT
~p I - tI 'f ° ’
(124)
”i” Cf ^3 ® •
(125)
dP = d(pr + pa) = Upr + Pa) ^ — Pa^y •
(l2())
Now
PHYSICAL PRINCIPLES
S7
Hence, (123) is the same as
(4pi- + />») y. + + Po) “ P»] Y ~ ° ■
From (122) and (127) wc find that
X 2 pr + ^ Pa A _1_ A
7-1 ^ Apr + P a
Apr + Pa y\ipr + Pa) ~ Pa
Let US now define the quantity jS as follows:
pP = pa *, (l 0)P — pr •
Equation ( r 28) can now be written as
12(7 - i)(t - g) + <3 _ 4 - .S /3
(7 - 0(4 - .Sl8) ■■ I'. - ■
Solving for P,, we find that
r. = /3 +
(4 - ,S/3)-'(7 - 0
fS + 12(7 - i)([ - IS) ■
From (r3i) we see that I’, = 7 when ft = \, and P, = 4/3
ft = o.
Again, from (124) and (126) we have
Apr + pu + J _ |i (pr + Pk) y* ~ Pu y — ° •
From (122), ( 132), and (120) we now have
i2(t -ft) + ^ ft
7 — r
(4 - ?,ft) +
4 - .3/3
ft
I -
Solving for P^., we find that
^ I (47 .S ^)(7 - t)
' '^^•' + .3(7- 0(i - «(4 + ^)’
S8 STUDY OF STELLAR STRUCTURE
From (127) and (132) we have
or
I +
I r.
4 - 3iS I - r.
(13s)
= (136)
Finally, we obtain the following equation, expressing Fj in terms
of Fx:
^ (4 - 3 i 8 )rx
^ 3(1 - , 3 )rx + 18 •
(137)
We see that when ^ = 1, F, = Fj = 7; and when /3 = o, F, =
r* = 4/3-
To determine Fj we proceed as follows: Eliminating dP/P be-
tween (123) and (124), we have
dT , (Fx - i)r. dV
T Fa V ~
(138)
Comparing this with (125), we have
(F, - i)F.
^ K •
(136) and (139) we find
r3 = I +
Fx - 18
4 - 318 ■
From (131) and (140) we finally have that
(4 — 3 i 8)(7 - i)
r, = I +
^ 12(7 - i)(r - (3) ■
(130)
(140)
(141)
Fj has the same limiting values for |8 = i and |3 = o as F, and F^.
Table i gives the values of the exponents F,, Fj, and Fj for different
values of j8 for a monatomic gas (7 = 5/3)-
PHYSICAL PRINCIPLES
TABLE 1
S9
l-p
r,
r.
Vi
I-/3
r,
r*
Ti
1.667
1.563
1. 511
1.476
1.449
1 .426
1 .667
1.484
1 .417
1 .667
1 .510
0.6
1.405
1-343
I- 3 S 9
0.7
1 .386
1-338
1.350
1 .444
0.8
1.368
1.335
1.344
0.2
1.383
1.363
1.351
1 .408
1.386
1.370
0.0
I -350
1.333
1.338
0-3
0.4
0.5
1.0
1.333
1.333
1.333
Equations (121) and (126) enable us to determine the specific
heats at constant volume and pressure for an inclosure containing
matter and radiation. Thus from (121) we have
or, in terms of jS,
CV = ‘'I [|3 + 12(7 - i)(i - ^)] • (h.O
P
Using equation we have alternatively
CV = evu - ^
Similarly, eliminating dV between (121) and (126) and putting
dP = o, we find that
C,, = [/3^ + (y - i)(4 - ^P)‘ + ^2(7 - ~ ^^)l > ('45)
or, using (i.^i),
CV = r.l^+ 12(7- i)(i - /3)1- ('4f>)
P
Equations (130) and (i.V») i^^nable us to express in the following
alternative forms:
_ (7 - 0(4 -
f r - CV
(7 — i)(4 —
0
(147)
I’rom (14,0 and (14^’) we find that
Cr
r.
/. .uN
6o
STUDY OF STELLAR STRUCTURE
BIBLIOGRAPHICAL NOTES
Poly tropic changes were first considered by G. Zeuner {Technische Thernto-
dynamik, i, Leipzig, 1887). The systematic use of a “polytropic” as a funda-
mental thermodynamical notion is, however, due to Emden. The results con-
tained in §§4, s, 6, and 8 are due to Emden. Reference should be made here
to his daissical treatise Gaskugeln^ Leipzig, 1907.
The uniform expansion and contraction of gaseous configurations were first
considered by A. Ritter {Wiedemann Annalen, 5, 543, 1878). The treatment
of the problem given in the text (§ 7) follows, more or less closely, Ritter’s
original treatment. Historical remarks concerning the association of a part of
the results of § 7 with the name of Homer Lane are made in the bibliographical
note for chapter iv.
The virial theorem proved in § 9 is due to H. Poincare (JLeQons sur Ics hy-
pothbses cosmogoniquesj § 74, Paris, 1911) and A. S. Eddington {M.N., 76, 528,
1916). The applications in the form given in § 10 are due to Poincare.
The expression Pi (Eq. [13 1]), for the adiabatic exponent for indosures con-
taining matter and radiation, is due to Eddington (M.iV., 79, 2, 1918); it has,
however, been generally overlooked that there are two other equally possible
definitions for the adiabatic exponent, namely, Pa and Pj. The expressions for
Cp and Cv (Eqs. [143] and [146]) are given here for the first time.
CHAPTER III
INTEGRAL THEOREMS ON THE
EQUILIBRIUM OF A STAR
As was emphasized in the Introduction, the structure of a star
depends on a multitude of variables, and an approach toward a
detailed theory is made only by introducing assumptions and ap-
proximations of various kinds with a view toward discriminating
between the relevant and the less relevant aspects of the physical
situation. It is therefore necessary to introduce one assumption at
a time and investigate how far we can proceed with one assumption
before we feel the need to make another. In this chapter we shall
be mainly concerned with an attempt to discover how far we can
proceed with the assumption that a star is in a steady state in gravi-
tational equilibrium. Wc shall supplement this further by the as-
sumption that the density distribution is such that the mean den-
sity pfr), interior to given point r inside the star, does not increase
outward from the center. We shall see that these two assumptions
already enable us to determine the order of magnitude of some of
the more important physical variables describing a star. The meth-
od consists in finding inecjualities for quantities like the central pres-
sure, mean i)ressure, the potential energy, the mean value of gravity,
etc. Before i)roceeding to establish the inequalities, however, we
shall obtain the ecjuations of equilibrium and some general formulae.
I. Equations of gravitational equilibrium. We shall be concerned
only with spherically symmetrical distributions of matter. Let r de-
note the radius vector, measured from the center of the configuration .
Since we have a si)herically symmetrical distribution of matter, the
total pressure the density p, and the other physical variables will
all be functions of r only. Let M{r) be the mass inclosed inside r,
'I'hen
M{r) = ^TK^pdr ; dM{r) = ^Tf^pdr , (i)
()L
62
STUDY OF STELLAR STRUCTURE
We shall denote by p(r) the mean density inside r, and by p the mean
density for the whole configuration:
Hr) =
M(r) .
(2)
where M is the mass of the configuration and R defines the radius of
the configuration at which p and P vanish.
Consider an infinitesimal cylinder at distance r from the center
of height dr, and of unit cross-section at right angles to r (see Fig. 4).
Let P be the pressure at r and let the in-
crement in P as we go from r to r + dr
be dP, The difference in pressure dP
represents a force, —dP, acting on the
element of mass considered, in the di-
rection of increasing r. This must be
counteracted by the gravitational attrac-
tion to which the element of mass is
subjected. The mass of the infinitesimal
cylinder considered is pdr. The force of
attraction between M(r) and pdr is, ac-
cording to elementary potential theory, the same as between a mass
M{r) at the center and pdr at r. By Newton’s law this attractive
force is given by GM{r)pdr/r^, where G is the constant of gravita-
tion. Further, the attraction due to the material outside r is zero.
Hence, for equilibrium we should have
iP .
r‘
(3)
dP GM{r)
dr ~
(4)
It should be noticed that we have used P to denote the total pres-
sure; thus, if we are considering a gaseous star, P is the sum of the
gas kinetic pressure and the radiation pressure (according to the
Stefan-Boltzmann law). We shall then write
P = pT + ,
(S)
THE EQUILIBRIUM OF A STAR
63
where k is the Boltzmann constant, m the mean molecular weight,
H the mass of the proton, and a the Stefan-Boltzmann constant
(chap, ii, §11). In (5) we have used {k/fj.H) in place of the gas
constant “J?,” as hitherto. This is more convenient, and in the fu-
ture we shall adopt this definition consistently.
Finally from (4) and (i) we have our fundamental equation of
equilibrium :
2. The poienlial and the potential energy . — The gravitational po-
tential V is defined as the function the derivative of which in a given
direction represents the gravitational attractive force in that di-
rection acting on unit mass. For a spherically symmetrical distribu-
tion of matter, V must be clearly such that
dV^GM^r) .
dr ‘ ^
h.qualion (4) can now be written as
I dr ^ _dV
p dr dr
(«)
If r ^ R, ^f(r) = A/ == constant; we can therefore integrate (7)
and obtain
(r^K). (9)
(Equation jc)! is so normalized that \/ = o as r ~> co .) In ]xirticu-
lar, the |)otential V, at the boundary is given by
= -
GM
R ‘
(10)
'fhe potential energy of a given distril)ulion of matter is delined
as the work done (on the system) to bring the matter “dilTused”
to infinity into the given dislril)iition. We shall denote the poten-
tial energy by 12. For a sjAierically symmetrical distribution of mat-
64
STUDY OF STELLAR STRUCTURE
ter, can be calculated as follows: Suppose that we have already
“brought” from infinity an amoxmt of material M{r). The work
done to bring an additional amount of matter dM{r) (as a spherical
shell of thickness dr) is
-GM{r)dM{r)
rt
Jr
GM(r)dM{r)
r
(ii)
Hence, the potential energy Q, of the configuration is given by
— r
M(,r)dMir)
r
(12)
Equation (12) is perfectly general, and is independent of the equa-
tion of hydrostatic equilibrium. For the case of hydrostatic equilib-
rium, equation (12) can be further transformed as follows:
or by (7)
dm
{M^{r)\^^dM{r)
(13)
= \G -k
(14)
2 R 2j
(is)
Again integrating by parts and using (10) for the value of V at
r = i?, we find that
/^R
= M VdMir)-, (16)
the important point to notice is the appearance of the factor 1/2
in (16).
We shall now proceed to establish a number of integral theorems
for configurations in gravitational equilibrium. The first three theo-
rems (due to Milne) are true for any equilibrium configuration;
THE EQUILIBRIUM OF A STAR 65
even the assumption ^^ip{r) does not increase outward” is not intro-
duced (the assumption is first made in Theorem 6).
3. Theorem i. — In any equilibrium configuration the function
GMHr)
decreases outward.
Proof: Equations (i) and (4) can be combined into
dP ^ _GM{r) dM{r)
dr 47rr‘* dr
Now,
d ^ ^dP ^ GM{r) dMfr) _ GM^{r)
dr I Stt;* ' dr /^irr^ dr
By (i 8 ), then,
d\ GM'ir)
dr 87rr*
GM^{r)
2Trr^
< o,
(17)
(18)
(19)
(20)
from which the theorem follows.
Corollary: If P,. denotes the central i)ressure, then wc should have
J\ > P +
GM\r)
87rr»
GM^
^ 87r/?» ’
(21)
The outer memlxTs of the foregoing inequality give
(/A/-
^ 87r/f» ’
(22)
or, inserting numerical values,
> 4.44 X ‘'yncs cni-^ ,
(23)
or
Pc > 4.50 X (^) >
(24)
where O and Rq refer to the mass and the radius of the sun.
66
STUDY OF STELLAR STRUCTURE
4. Theorem 2. — For any equilibrium configuration
if
V < A •
From (18) and the definition of I, we have
rdp ,
Ir^
( 25 )
(26) *
(27)
or, integrating by parts and remembering that y < 4, we have
Ip = 4ir(4 — v) ^ Pr^~'’ dr , (28)
which proves the theorem.
We notice that when y = 4, (27) can be integrated, and we have
= 4irP,. .
(29)
Again, by (12)
(30)
and hence, by the theorem
— == 1 2^^ Pr^dr ,
(31)^
or
-12 = sj'^'pdV ,
(32)
where dV stands for the volume element.
* Actually, we shall sec (§ ti) that under “normal” circumstances the integral Iv
converges for v < 6 ,
“ Equation (31) was known to A. Ritter {Wiedemann Annalcn, 8, 160, 1879).
THE EQUILIBRIUM OF A STAR
67
Now the value of the gravity g at r is clearly GM{r)li^. Hence,
if we denote by g the mean value of gravity defined by
Mg
then
-_(■%«« = (33)
= Prdr . ( 34 )
Mg
5 . 1 'heorem 3 . — For any equilibritm configuration
„ „ ,4-1' CM^ ^ ^ ^ GM‘
(35)
if
V < 4 .
(36)
Proof: Hy 'rheorem 2
Iv = 47 r (4 — v)^ Pr'^~^'(lr .
(37)
Hut l)y Thoorom r
CM^ _ GM^{r) _ GM^ir)
<S7r/e‘ 87rr' ‘
(38)
By (37) and (3.S) we have
47 r( 4 -.')(
8 ,rri
or
>
'' > Av +
f-y f'(Jr ,
(M))
Hr)(lr (fA/“
^ 2 /e'* '
(40)
Now,
nau^ir
Jo
dr = - l£ C.Mfr) £l£r , (40
68
STUDY OF STELLAR STRUCTURE
or, after an integration by parts,
P GM*{r) ^ GM^
1 r'+>' vR>>
2 GM{r)dM{r)
^ vjo r”
( 42 )
GM’
_2. 7
( 43 )
vR- ■'
- Iv •
V
Inserting (43) in (40), we have
4irPJ?4->' > 7, + I, - ^ ,
V 2v R' 2/?"
or, simplifying, we have
which proves the theorem.
Corollary i: If = i, /i = — Q, we have
(44)
(45)
(46)
That GM’‘/ 2 R sets the absolute minimum to — S 2 was first proved
by Ritter {Wiedemann Annalen, 16, 183, 1882).
Corollary z: If y = 2, Jj = Jl^fg, and wc have
2TePcR^ "[■
I GM^
4
> Mg >
GM^
2 R^
(47)
The following theorem is due to Ritter.
6. Theorem: 4. — In a gaseous configuration m equilibrium in which
the radiation pressure is negligible.
= ^ I ^ GM
6 k R ’
(48)
where the mean temperature T is defined by
MT = CxdMir)-,
(49)
fi is further assumed to be constant in the configuration.
69
THE EQUn.IBRIUM OF A STAR
Proof: If the radiation pressure can be neglected,
0 * r r -
P-^pT, 0,
(so)
Hence,
11
II
>3
(si)
V 0
II
(S2)
By (.32), then.
M?. -■'''a.
3 *
(S 3 )
By Theorem 3, Corollary i, we have
= I Mf/ C'M
^ ^ 6 k R ’
(S 4 )
thus proving the theorem.
Inserting numerical values in (54), we find that
f > 3.84 X 10" M Q ;
(54')
in other words, we may expect the temperature to l)e of the order
of a few million degrees in stellar interiors.
Equation (5,0, derived above, has an important physical mean-
ing. If we are considering a gaseous configuration (and if we neglect
radiation iiressure), then the internal energy is given by
V = n- /7V/M(r) = cvMT
70 STUDY OF STELLAR STRUCTURE
a formula which was derived independently by A. Ritter and J.
Perry.’ (We shall refer to [55] as “Ritter’s relation.”) We have al-
ready derived (55) from the virial theorem (chap, ii, § 10).
7. Theorem 5 . — is the mean pressure interior to r, defined by
Mir)P(r) = CrdM^r) , (S6)
then in any equilibrium configuration
P{r) - P(r) > . (^ > o) . (57)
12T
Proof: Integrating by parts the integral defining P{r), we have
M{r)^{r) - P(r)] = - J^V(r) JP , (S8)
or, using (18),
M{r)[P{r) — P(r)]
- r
I27r Jo
d[M^{r)]
r*
( 59 )
or, again integrating by parts,
M{t)[P{t) - P(r)] =
G mr),G
i27r r^ "^ 37 *' Jo
(60)
Since the second term on the right-hand side of (60) is positive, we
have the inequality stated.
Corollary: If we put r = P in (57), we have for the mean pres-
sure P defined for the whole configuration the inequality
P >
I GM‘
12T
(61)
3 A. Ritter, op. ciL, pp. 160-162; J. Perry, Nature, 60, 247, 1899. Lord Kelvin, in
his work (referred to in greater detail in chap, iv), refers to (55) as the “Ritter-Perry
theorem.”
THE EQUILIBRIUM OF A STAR 71
8. 'Pheorem 6. — In any equilibrium configuration in which the
mean density p(r) interior to r does not increase outward^ we have
^ Pr. — F ^ , (62)
where pc is the central density.
Proof: From (18) we have
r> ('MirJd-Mij)
_p^G rM{r)c
4irJo r
From the delinition of the mean density p{r) (Eq. [2]), we have
=
I n’rp
Mir)
i’tpW.
Insertinji (64) in we have
4’r X
P‘\t^(r)M {r)dM(r) . (65)
Since by hyi^othcsis p(r) docs not increase outward, we have from
(O5) that
l\-r^ ' ('M'f^ir)dMir) . (66)
47r Ja
'The inlcf^ral on the right-hand side of (Of)) can be evaluated, and
we havt‘
Pr ~ r .Uil7r)'^'-<(;p«/K0A/^^K0 • ( 67 )
Again, from (05), according to our hy])olhesis,
P. - P ■<, ' ('a 1 '/■<{r)dMir) , (68)
4^ Jo
Pr - P .
Combining (O7) and [in)), we have the recjuired inequality.
72 STUDY OF STEULAR STRUCTURE
CoroUary: If we put r = Rin the inequality of Theorem 6, wc
obtain
^ P. ^ . (70)
In the left-hand side of the inequality we can substitute for p its
expression We then find
£ ^ i’c < . (71)
We see that the additional restriction imposed on the density
distribution, namely, that p(r) does not increase outward, enables
us to improve the inequality obtained for Pc in Theorem i. Nu-
merically we now have that
■Po ^ I -35 X atmospheres . (72)
Equation (71) was first given by Eddington,^ but the complete theo-
rem and the proof given are due to Chandrasekhar. Milne has given
the following instructive alternative proof for the inequality
Consider the expression
p ^ 2 . C.M‘
^ ° ^ 8t •
P+ a
GMKr)
Sirr-t ’
(73)
(74)
where a is, for the present, an arbitrary number. Now,
fp j. _L
dr * J dr * 4irr‘' dr
GM^{r)
2wr^ ’
(75)
or by (18)
d
dr
R + a
GM‘(ry
Srrr^
-(a
. (if _ GM^(r)
^ dr ^ 2Tcr^ ^
(76)
4 Eddington stated the result only for p (not p(r)) decreasing outward.
THE EQUILIBRIUM OF A STAR
73
or, again, by (4)
d
dr
[(-■>-« 3 ]. (n)
or by (2)
il-
drL'
P+a
GMKr)
(,8)
If the mean density decreases outward, it is clear that p(r) ^ p.
Hence, if we choose a = 3, we have
d
dr
P + ^
GM^{r)
= ~ 2 [p(r) - p 1 ^ o . (79)
Hence, the expression (74) considered with a = 3 is a decreasing
function of r:
^ P +
3 GM^{r)
Stt r*
> 3
^ Htt ’
(80)
thus establishing the inequality. Milne’s i)roof cannot, however, be
extended to give the coni])lete 'nieorem 6.
9. TirEOKicM 7. The ratio (r — of lire radiation pressure to
the total pressure at the center of a wholly ifascoiis cofifi^uralion in
cquilihrium in %vhicli p(r) does not increase outward, satisfies the in-
equality
T - jSJ,. ^ T - , (81)
cohere s<itislies the quartic equation
M =
r/a j
(82)
is the mean molecular weight at the center.
Proof: Now, according to (5), the total i)ressure P is given by
P =
ixll
pT + laT^ .
(83)
74 STUDY OF STELLAR STRUCTURE
Define the quantity (i — j 9 ) by
(i - P)P = iaT* ; pP^-^pT.
( 84 )
By ( 84 )
I 1 k ^
. 1-03^
(8S)
or
T = [A.3-izi_^rV3
(86)
Again,
p = l±pT= [('i- Y 5 p4/3
ppH” LW/ a j34 J ■
( 87 )
Hence, at the center of the configuration
By Theorem 6, on the other hand,
■P,, ^ . (8g)
Comparing (88) and (8g), we have
Defining (i — jS*) as in equation (82), we have
I - /3* > I - |3„
/3*' ^
(92)
But (i — j 8 )// 3 ^ is a monotonic increasing function of (i — ^).
Hence, we should have
I - 0* ^ 1 - ( 93 )
which proves the theorem.
THE EQUILIBRIUM OF A STAR 7 S
10. Theorem 7 (which is due to Chandrasekhar) shows that for
a gaseous star the value of (i - / 3 ) at the center cannot exceed an
amount depending on the mass of the star only. Table 2 gives the
value of (i — |8*) for different values of the mass M.
TABLH 2
Values of (i-jS*)
As an example of the application of 'Fable 2, we see that for the sun
(i _ |8„) < 0.03 while for Capella {M = 4.18O), (i - / 3 «) < 0.22,
assuming in both the cases that /*« = , , ,,
II. 'Fheouem 2 ,.- For L, defined as in Theorem 2, and under ihe
conditions of Theorem 6 , we have
3 aM‘ ^ r ^ 3
f, 1 f (y- r rS
{v < 6 ), ( 94 )
= dUY'' r
Since p(r) does not increase outward, the minimum value for tlm
integral on the right-hand side is obtained by replacing p(r) by p,
and taking it outside the integral sign. Similarly , the maximum value
76
STUDY. OF STELLAR STRUCTURE
is obtained by replacing p(r) by pc, and taking it outside the integral
sign. In this way we find that
But by definition,
|«ip« = M = friZsp . (99)
Usmg (99), (98) is found to reduce to
3 GM« 3 GJ17*
which proves the theorem.
Incidentally, we have also proved that the integral defining 7„
converges for v < 6 if the mean density decreases outward and if fur-
ther is finite.
Corollary i: If p = i, 7i = — Q, and we have
„ o =5; . ( EOl )
Corollary 2: If p = 2, = Mg, and we have
zGM :^GM
4 J?»' ^ ^ ^ 4 r\ ■
(loiO
Corollary 3: If we put p = 6 on the right hand side of (97) and
extend the range of integration from r = r to r = 72 , we find
.Gilf> M . .
j| 7 (V) •
12. Theorem g.—In a gaseous configuration in equilibrium in
which the^ radiation pressure is negligible and in which, further, p(r)
does not increase outward,
= X tiH GM
S k Ta ^ '5 k R ’ (^° 3 )
77
THE EQUILIBRIUM OF A STAR
where y, {the molecular weight) is assumed to be constant in the configu-
ration, and T is the mean temperature defined as in Theorem 4.
Proof: By (53) we have for the case considered
MT = - ^ 2 . (104)
3 «
By Corollary i of the last thc'orem we have
5 ^ S
(los)
Combining (104) and (105), we have the required inequality.
We are thus able to replace the “1/6” that occurred in Theorem 4
by “i/s” because of our additional hypothesis concerning p(f). Nu-
merically (103) reduces to
7' ^ 4.61 X loV
M Rq
O R
(106)
13. I'hicorem 10. If I,,,, is the integral defined by
j„„ = j" [3(<T +!)>>']. (107)
then under the conditions of Theorem 6
, (.08)
30- + ^ — V R - 3°’ + 3 “
where rc is defined as in Theorem S
Proof: Since
1 MW
r =
LS’tK'')!
(109)
we have from (107) that
la,, = CdUyl^^fp''Kr)M^'’-'^IKr)dM{^ .
Arguing as in Theorem 8, wc easily obtain the inequality (108).
(no)
78
STUDY OF STELLAR STRUCTURE
14. Theorem ii. — If P is the mean pressure defined by
MP = J'* PdM{r) ,
(hi)
then, under the conditions oj Theorem 6 ,
3 GM* p '' 8 GM^
207 r rj ^ 2 ot
(112)
Proof: By definition
MP = f^PdMir) ,
(113)
or, integrating by parts,
MP = - ^^M{r)dP .
(114)
Since (Eq. [i 8 ])
(IIS)
we can re-write (114) as
jlfp _ G M‘{r)dM(r)
4 ’rjo »•“ ’
(116)
or, in terms of the integral „ introduced in Theorem lo,
we have
MP = - - I, 4 .
47 r ’
(117)
By Theorem 10 we then have
9 .M 1 > p > _ 3 _
207 r rj 207 r ’
(118)
which proves the theorem.
Numerically we have
^ S -4 X (^) *
(119)
79
THE EQUILIBRIUM OF A STAR
15. The physical content of Theorems 6, 8, 9, 10, and ii is the
following ; We are given a certain equilibrium configuration of mass
M and radius R with some arbitrary density distribution, arbitrary
except for the condition that the density p(r) does not increase out-
ward. From the given configuration we can construct two other
configurations of uniform density one with a constant density
equal to p, and the other with a constant density equal to p, (see
Fig. 5). 'rhe radii of these two configurations are clearly R and Tc,
R R
Ki(!.
respectively. 'I’heorcms (), 8, q, 10, and ii simply state that the
lihysieal variables characterizing the given equilibrium configura-
tion, namely, P,-, -il, H, T (for the case of negligible radiation pres-
sure), and P, have values respectively less than those for the con-
figuration of uniform density with p = p,., and respectively greater
than those for the configuration of uniform density with p = p.
Thus, the givcMi conliguration is, in this sense, intermediate between
the two configurations of uniform density with p = Pc and p = p,
resi)ectively.
16. 'I’liicoRiCM 1 2. Under l/w condUions of Theorem 6 wc have
[>rm'ided
0 > K ^ 4 ,
cohere (120) is a sir id incqualily Jor »» > 4.
Proof: C'onsider (he integral /„:
G M{r) dM (r)
(120)
(121)
(122)
r‘
8 o
STUDY OF STELLAR STRUCTURE
By (i8) we can transform this into
r r
1 , = 4T 1 - .
Jo
(123)
Since we have assumed that ^ 4, we clearly have
(124)
In (124) we have the equality sign for the case v = 4.. For v > 4
we have a strict inequality. On the other hand, by Theorem 8 (Eq.
[98]) we have
I- (|Tp,)'/3GJlf »->■)/•' {v
< 6) . (125)
Combining (124) and (125), We have
(126)
or
0 — V
(127)
Again, (127) is a strict inequality for v > This proves the theo-
rem.
If we write
(128)
equation (127) reduces to
yc
if
I < w ^ 3 .
(129)
(130)
Further, Sn introduced in (122) stands for the numerical coefficient
(131)
= (It)'/’
THE EQUILIBRIUM OF A STAR 8i
Finally, (129) is a strict inequality for w < 3. Equations (129) and
(131) bring out clearly the critical nature of w = i and = 3, a
circumstance we shall again encounter in the future. Table 3 gives
the numerical values of Sn for different values of n.
TABLE 3
Values of Sn
n
‘S’n
n
^ .0
0.806
2.0
1.364
2 .S 99
2.5
0.98s
1.5
17. Homologous transformaimts. — A general homologous iransfor-
malion is one in which the densily and the linear dimensions at each
point are multiplied by constant factors to obtain another equilibrium
configuration,
'Jhe general homologous transformation is best considered as
l)eing “built” up of two elementary homologous transformations:
(a) the transformation in which the radial dimensions are kept un-
altered while the density at each i)oint is multiplied by a constant
factor .v; (/;) the transformation in which the configuration is sub-
mitted to a uniform expansion or contraction (in the sense already
defined in chap, ii, § 7) when the radial dimensions are altered in
the ratio i :y.
We shall ])rove the following theorem:
Tukorkm 13. JJ the radiation pressure is a fraction (i — j8„) of
the total pressure at a given poinl in an equilibrium conjiguralion, and
if it is a fraction ( i — Pi) at the corresponding point in a homologonsly
transformed configuration, then
where M „ and M , refer to the ina,ss of the configuration before and after
the homologous transformation.
Proof: We shall consider the homologous transformation as built
up of two elementary homologous transformations, as already ex-
plained.
82
STUDY OF STELIAR STRUCTURE
First let us consider the homologous transformation in which the
radial dimensions are unaltered. Then the density, p, and the mass
interior to r, M{r), are each multiplied by x. From the equation
dr
(133)
we see that P gets multiplied by a:®. But according to equation (87),
• = \( ^ Vs I -
a j84 J
>4/3
(134)
Hence, the left-hand s^ of the foregoing equation gets multiplied
by while the right-hand side gets multiplied by Hence,
the term involving j 3 must get multiplied by In other words,
I - /3_i ^ I - jSo 3 ^ I - iSo / M^y
nWi uiiPt \Mo) ’
(13s)
Now, consider a uniform expansion, in which the linear dimensions
are increased in the ratio i:y. As shown in chapter ii, § 7, the effect
of this transformation is to multiply P and p by y~^ and y~\ re-
spectively. From (134) we now see that for this transformation the
left-hand side gets multiplied by while p^/^ on the right-hand
side also gets multiplied by Hence, (i — i8)/j8V is invariant
to this transformation. Hence, (13 s) is true for a general homologous
transformation.
The foregoing theorem is of importance in the theory of gaseous
stars in so far as it shows that, if we consider a sequence of homolo-
gous gaseous configurations in equilibrium, then the relative impor-
tance of the radiation pressure — as measured by i — jS — increases in
the direction of increasing mass along the sequence.
BIBLIOGRAPHICAL NOTES
Integral theorems have been considered by Ritter, Eddington, Milne, and
Chandrasekhar.
A. Ritter, Widcmami Annalenj i6, 183, 1882. The inequality for Q (Theo-
rem 3, Corollary i) and Theorem 4 are proved here.
A. S. Eddington, The Internal Constitution of the Stars j pp. 90-94, Cam-
THE EQUILIBRIUM OF A STAR
83
bridge, England. The inequality (73) is stated (and proved by means of physi-
cal arguments). Theorem 9 is also considered, though the proof given in the
text is different from Eddington^s.
E. A. Milne, M.iV., 89, 739, 1929; ibid., 96, 179, 1936. Theorems i, 2,
and 3 are proved here, while parts of Theorems 6 and 8 are proved by different
methods.
S. Chandrasekhar, M.N., 96, 644, 1936; Ap. /., 85, 372, 1937. Theorems 6,
7, 8, 10, II, and 12 are proved in these papers. Theorem 5 is proved here for
the first time.
CHAPTER IV
POLYTROPIC AND ISOTHERMAL GAS SPHERES
In the last chapter we considered the most general properties of
equilibrium configurations. In this chapter we shall be concerned
with the detailed study of a class of equilibrium configurations re-
sulting from a special kind of relation between P and p. Formally,
the fundamental problem is the study of equilibrium configurations
in which P and p are connected by a relation of the kind
p = a:p<»+'^/” , (i)
where K and n are constants. This problem, toward the solution of
which fundamental contributions have been made by Lane, Ritter,
Kelvin, Emden, and Fowler, is also of considerable physical interest.
We shall, therefore, first consider the physical circumstances which
led initially to the study of the equilibrium configurations with an
underlying “equation of state” of the kind (i).
I. Convective and polytropic equilibrium . — ^The physical notion of
convective equilibrium was first introduced by Lord Kelvin in 1862
in connection with some of his considerations relating to the tem-
perature of the earth’s atmosphere.' Kelvin defined convective equi-
librium in the following terms:
Any fluid under the influence of gravity is said to be in convective equilibrium
if the density and the temperature are so distributed throughout the whole
fluid mass that the surfaces of equal density and of equal temperature remain
unchanged when currents are produced in it by any disturbing influence so
gentle that changes of pressure due to inertia of motions are negligible.^
Kelvin further comments that
the essence of convective equilibrium is that if a small spherical or cubic portion
of the fluid in any position, P, is ideally enclosed in a sheath impermeable to
‘ Sir W. Thomson (Lord Kelvin), Mathematical and Physical Papers, 3, 25S“26o,
Cambridge, 1911.
’ Ibid., 5, 2S4“283. The quotation is from p. 256.
84
POLYTROPIC AND ISOTHERMAL GAS SPHERES 85
heat and expanded or contracted to the density of the fluid at any other place
P', its temperature will.be altered, by the expansion or contraction, from the
temperature which it had at P to the actual temperature of the fluid at P'.
It is clear that the process considered is a quasi-statical adiabatic
change, and consequently the equations to be used are (Eq. [19], ii)
p = constant • = constant ; Tp^~^ = constant , (2)
where 7 is the ratio of the specific heats. It is seen that the relation
connecting p and p is of the form (i).
The gravitational equilibrium of a gaseous configuration in which
p and p arc related as in (2) was first considered by Lane^ (1870),
but the same problem was independently considered by Ritter^
(1878) and also by Kelvin'*^ (1887).
In applying the equations (2) of adiabatic expansion or contrac-
tion to a spherical mass of gas in convective equilibrium, Kelvin^’
makes the following interesting remarks:
If a gas is enclosed in a rigid spherical shell impermeable to heat and left
to itself for a sullicicntly long time, it settles into the condition of gross-thermal
equilibrium by “con(luctit>n of heat” till the temperature becomes unifomi
throughout. Hut if it were stirred artificially all through its volume, currents
not considerably disturbing the static <listribiition of pressure and density will
bring it approximately to what I have called convective equilibrium of tempera-
ture. 'riic naliird siirruifi produced in a great free (luid mass like the Sun’s by
the cooling at the surface, must, 1 believe, maintain a somewhat close approxi-
mation to convective cfiuilibrium throughout the whole mass.
It follows from Kelvin’s remarks that we are entitled to use the equa-
tions (2) for an adiabatic expansion or contraction provided that
during the process of “stirring** the appropriate dQ = o. But this
need not in general be the case. Indeed, in his very first application
of the idea of convective equilibrium (to the earth’s atmosphere,
with a view to calculate the fall of temperature with height), Kelvin
had to consider the case where the “stirring” led to a physical proc-
ess in which dQ 9^ o. The difliculty arises from the circumstance
3 J. IlonuT Lane, Awer. J. .Vd., 2d ser., 50, 57, 1870.
•» A. Ritter, WinlrnKmn A umlcnt 6, 135, 1S78.
s W. Thomson, Phil. Mag., 22, 2S7, 1887.
^Ibid, Also Thomson’s Collccicd Papers^ 5, 184-190. 'J'he quotation appears on
p. 186.
86
STUDY OF STELLAR STRUCTURE
that if we consider the ‘‘natural stirring” of a moist atmosphere the
condensation of vapor in the upward currents of air is of considerable
importance. This latter problem was also considered by Kelvin
(at Joule’s suggestion) and is, of course, of fundamental impor-
tance in meteorology. In modern versions’ of Kelvin’s work such
changes are generally considered to be represented by the equation
dQ = cdTj where c is taken to be approximately constant. More
generally, if, during the process of stirring, the quantity of heat,
dQj supplied is proportional to the instantaneous change of tempera-
ture, dT, then dQ = cdT; this is the definition of a polytropic
change. We then have
p = constant * ; 7' = . (3)
cy — c
Hence, the consideration of polytropic changes is more general
than the consideration of adiabatic changes; the latter is obtained
as a special case when we put c — o. For this reason Emden con-
sidered polytropic-convective equilibrium.*
If we use the variables introduced in chapter ii, § 5, we can
write
p = ; P = ^ e^,X‘»+'’/nen+i ; n = » (4)
where 6y is the polytropic temperature, which is, of course, the
same for all parts of the gaseous sphere. For the adiabatic-convcc-
tive case 7' = 7 and 9 ^, is the adiabatic temperature. Since in all
these considerations radiation pressure has been neglected, we can
write
P = A'p‘+C/’‘) ; (s)
We are thus led to consider the mathematical problem of deter-
mining the structure of an equilibrium configuration in which P
and p are related according to equation (i); when we wish to con-
7 See L. Weickmann, “Mechanik und Thermodynamik der Atmosphare,” in Lvhr-
buch dcr Gcophysik herausgegeben von B. Guttenberg, pp. 797-965 (Berlin, 1929).
® K. Schwarzschild, Vierteljahrsschrifi dcr astron-omischen Gescllschaft, 43, 26, 1908,
POLYTROPIC AND ISOTHERMAL GAS SPHERES 89
3. Transformations of the Lane-Emden equation . —
a) Put
*=!■
The equation (ii) easily reduces to
_ x"
(13)
(14)
h) Kelvin’s transformation . — ^Instead of f , introduce the new vari-
able X defined by
X =
r
(15)
The Lanc-Emden equation now transforms into
dx*
(16)
c) The singular solution for n > 3. — We first ascertain whether
(16) has a solution of the form
0 = ax^ (17)
for a suitably chosen a and Substituting (17) in (t6), we have
a(b{d 3 — , (18)
an equation which must be valid for all values of x. Hence, we
should have
or
w 4" 2 = ncj ; * = w(i — w) ,
(19)
a
2{n - 3)
(» - 0'
(20)
For n > 3, and <o < i we therefore have the singular solution
0 . =
2( n — 3) '
(«■- x)\
a ,
(21)
90
STUDY OF STELLAR STRUCTURE
or, in' terms of
ja/(n-i) •
( 22 )
For n ^ ^ we have no proper singular solution of the type (17).
d) Emden^s transformations . — Since (17) defines a solution of (16)
(for n> 3), we make the substitution
6 = Ax^z ; (h = ,
’ n — 1
(23)
where A is, for the present, an arbitrary constant, which we shall,
specify later. From (23) we obtain
dx^
. x^
- d^z , - - _ dz , _ / _ \ -
(24)
Substituting (24) in (16) and using the relation w + 2 = nu, wc
have
dx^
- + 2coa: ^ + w(« — i)z + =
( 2 S)
We can eliminate x from the foregoing equation by making the fur-
ther substitution
X = f = log a: = — log 5 . (26)
From (26) we easily find that
dz ,dz d^z dz~\ . .
Tx-^Tt’
Substituting (27) in (25), we obtain
d^z dz
^ + (2W - i) ^ + <i(w - l)z + = o . (28)
We shall consider two forms of (28) :
Case i: n > 3. — For w > 3 the singular solution (21) is proper,
and we shall therefore choose A — a. By (19),
A^~^ = = «(i — w) (w > 3, CO < i) . (29)
POLYTROPIC AND ISOTHERMAL GAS SPHERES 91
Equation (28) now takes the forni
+ (2« - l) - i(l - - Z”-0 = ° >
(30)
dt^ ' ■'
or, since w = 2/(» — i), we have
d‘z , 5 - nclz _ 2 {h - 3) / _ 2..-.) = o . (31)
dF^ir-ldi {n-iy
The singular solution (21) is delined by s = i.
Case a. We choose yl = i. Our equation then is
d‘z . S ~ 'iE 4. z + z” = o .
Jf. + dl ^ (« - i)'
(32)
4. The. iMnc-Emlcnfimlumsfor n = o, i, and 5.- We shall con-
sider these three cases separately.
Case, i: n = o.- The Lane-Emden equation is
(33)
which, after a first integration, yields
^.^1= (34)
^l,,re is an integration constant. A second integration now
yields
0 = 7; -P I -
(35)
where D is a second integration constant.
We see that the general solution of (.33) has a singular! y a
origin, and that
(? -> o) . (36)
0 ■'
If, however, we restrict ourselves to solutions which are Unite at the
origin, then C = o and
(37)
0 = 1) - ie •
92
STUDY OF STELLAR STRUCTURE
The Lane-Emden function is characterized by = i at the origin,
and hence the Lane-Emden function da is given by
8 a = x-^e- (38)
The function do has its first zero at ^ where
= V6 ; = o . (39)
Case ii: n = i. — Consider the Lane-Emden equation in the form
(14). Then, for « = i
The general solution of (40) is
X = C sin (J - 5) ,
where Cand S are constants of integration. By (13),
_ C sin (g - &)
(41)
(42)
If 5 o, the general solution has a singularity at the origin :
constant
o) . (43)
Again, if we restrict ourselves to solutions which are finite at the
origin, S = o and
_ C sin ^
r~
(44)
The solutions in the foregoing forms were first given by Ritter. The
Lane-Emden function 0, is given by
0 . =
sin ^
1 “'
(45)
The foregoing function has its first zero at $ = x and is mono-
tonically decreasing in the interval (o, x).
POLYTROPIC AND ISOTHERMAL GAS SPHERES 93
Case in: n = 5.— We shall consider the equation in the form
(31) with » = 5- We have
where out variables s and I are, according to equations (aa), (ag),
and (26),
i = ^ = e-« ; 0 = = iWy'^z . (47)
Multiplying both sides of (46) by dz/dt, we have
which can be integrated as it stands;
where D is a constant of integration. If + », then according
to (49), {dz/diy-*-^, and this is impossible since dz/dl is real.
We can therefore write
‘!l ^ = dl , (so)
±[2])+ is" -
and s can at most oscillate to and fro between the greatest and the
least roots of
2D + is" — I'js'’ = o . t.'iO
'I'he integration of (50) for non/acro I) is complicated and involves
elliptic integrals. The case of interest is, however, when I) - o.
Then,
= -idi, (52)
s(l - iz’')'/"
where the ambiguous sign has been so chosen that f » . Make
the substitution
IjS" = sin" f ,
94
STUDY OF STELLAR STRUCTURE
from which we have
and (52) becomes
dz cos f
— = 2 ^
z sin f
cosec fdf ^ — dt y
(54)
(55)
which can be integrated. We obtain
tan if = Ce ~-^ , (56)
where C is a constant of integration. From (53) we now have
1-1 _ 4tan4f
“ (i + tan« Uy ’
(57)
or from (56)
r i 2 cye-“
^ * [(i +C’c-»0“J •
(58)
By (47). then,
[{1 + oey •
(59)
The Lane-Emdeii function ^5 is therefore given by
which was independently discovered by Schuster and Emden. We
see that 65 is a decreasing function and tends to zero only as ^ > 00 ,
which means that the corresponding equilibrium configuration ex-
tends to infinity.
5. The Lane-Emdcn functions for general n. — ^Wc have seen that
the Lane-Emdcn function can be explicitly given for w = o, i, and 5.
Such explicit expressions for other values of n do not seem to exist,
and recourse must be had to numerical methods. A method of
constructing the Lane-Emden function would be to start with a
series expansion near the origin. We assume a series of the form
(61)
0 = I + +
POLYTROPIC AND ISOTHERMAL GAS SPHERES 95
The scries is thus chosen in order that the boundary conditions (12)
be satisfied; there can clearly be no term in since has to
vanish at the origin and consequently the series can contain only
terms of even powers in By substituting the foregoing series in
the Lanc-Emden equation and equating the coefficients of like pow-
ers in we can successively determine the coefficients c, d,
Thus, the series including the first three terms is found to be
= I - + (62)
By taking a sufficient number of terms in such a scries we can cal-
culate the values of 0 for ^ < i to any required degree of accuracy.
For $ > 1 the solution can then be continued by means of standard
numerical methods.
'rhe solution so constructed monotonically decreases from the
center, and for n < s has a zero for some finite ? = (say). At
$ = ^ has its first zero, and thus the configuration has a definite
boundary. As we have already seen for w = 5, the configuration
extends to infinity; the same is true for n > 5, as we shall sec in
§ 20.
'fables of the Lane-Kmden functions, On, are given in Emden’s
book ( Kjoy) for the values of n = 0.5, 1, 1.5, 2, 2.5, 4, 4.5, 4.9,
and 0. 'Tables of these functions were also computed by (i. Green
( i()0(S) for n = 1 .5, 2.5, and 4; these tables formed an appendix to
a ])aj)er by Lord Kelvin. Recently these functions have been com-
puted very accurately for //. == 1.5, 2, 2.5, 3, ,p5, 4, and 4.5 by D. H.
Sadler and J. (\ P. Miller.
'Table 4 gives the values of and of certain other functions (in-
volving (lO/il^ and $) at which are of interest.
6. Physical rliaraclcrisUcs. We thus see that if the Lane-limden
function is known, then we can construct for a fixed value of K (i.e.,
for a given ])olylropic temt)eratiire if we are considering convective-
polytropic efjuilibrium) a one-parametric family of configurations
by allowing X to vary continuously. Before we [proceed to show how
the Lane-handen functions are to be used in practice, wc shall first
derive some necessary formulae.
The Constants of the lane-Emden functions*
_ _ o
rf-oS VI ob Q O
00 tH O^Tj‘t>. 0'0 H o
irjso W so to m r
5 00 O' H VO CO »0 C
OOOOOOOOhm coo
Ov '^OO O VO
) CM vO H w O'©
V W w O 00 O' O
O' O v> OvoO »0 (O _
w VO Ov CO O 'OVO 0 w O' 8
<N CO t^vO O' O CO O' »0 M VO
0 O
OOwcowOOt^c^co
H « T*- rl- c<
N O'
6
teq
W PI 10 0 -^OO Tf 0 00
vO Plt^voO'O'O N O'
• 0 vO ^ M COOO H 0
. t>. CO N VO V )\0 0 lO ^ Q
• MvO '< 1 -cocOcOcO'^ Tj-vO CO 0
• PiOOOOOOOOOM
A
T
II
. M vO
(O'© ■ cs tJ-O © 0 10 O'
CO'O •COO©'NHMCSfOCO
CO H • 10© 00 O' "Ct N
cOPi •cOO'CSHTfOcsO'^
CO 0 - CO -^OO 0 V) c< CO M
00 •c^OcOWhhOOOO
• CO M
• M
JO.
I .0000
1.8361
3.28987
5-99071
11.40254
23.40646
54-1825
88.153
152.884
622.408
6189.47
934800
00
II
'VVK
O'© 'O 0 't 0 © CO 0 'O
00 M 10 0 0 p» PI 00 10 CN 00 >0 0
CO I'-' M Th M I >..00 O' 0 »-• J'- 'O PI
OVOO -rhM MOO H ^O'O'COCOCO
00 M 1-^ -"t H 0 O'OO C>. 1-^ t>.
1
tJ-cocOpipipipihmmhhh
O' >0 P >00 »0 Tf M 10©
tJ-OO t-OO PI 00 O'OO lO
O' f'l M CO PI 10© 00 10 M ©
ri- 10 '« 4 - 10 >0 lO O' H CO t-'* CO 0
-d- M © CO COOO 0 'O O'OO 0
PI PI CO CO Th irj© 00 O' M O'
M CO©
M
«
«o
II
e
• *o O 'O O 'O O IN lo O V) O' O
O O mm pi pi cococo'^tI*'^»0
•The values for n = 0.5 and 4.9 are computed from Emden’s integrations of for n = 3.25 an unpublished integration by Chandrasekhar has been used,
corresponds to the Schuster-Emden integral. For the other values of n the British Association Tables ^ \ ol. II, has been used.
POLYTROPIC AND ISOTHERMAL GAS SPHERES 97
a) Radius— The radius R of the star is given by (cf. Eq[io]) ;
, (63)
where defines the first zero of 0„.
The value » = i is a critical case, for if » = i, Si = ^'^d we
have , .
R = [Jg] -
which is independent of X. Hence, the radius of a polytrope of in-
dex I depends only on K, and is independent of the central density X.
If we are considering a configuration in convcctive-polytropic equi-
librium, the result shows that the radius of a polytrope of index i
depends only on its polytropic temperature.
Further, it is clear that for « = 5. ^11 finite values of X.
b) The mass relation. - -'I'he mass M(.^) interior to ? is given by
(65)
( 66 )
(67)
( 68 )
or, using (11),
or
M{i) = ^Tcpr^dr = 47ra‘^X i'
M(^) =
. do
MiO = -4’rtt’XS' ^ ;
substituting for a (Eq. [10]), we have
'I’he total mass, M, of the configuration is given by
'I'he value « = 3 is also a critical case, for, when n = 3,
(69)
98
STUDY OF STELLAR STRUCTURE
We thus see that the mass of the configuration depends only on K
and is independent of \. For the convective-polytropic case this
shows that the mass of a polytrope of index 3 depends only on its
polytropic temperature.
We notice, further, that when » = s the mass is finite, though
the configuration extends to infinity, for, according to the Schuster-
Emden expression for 6 ^, we find that
lim (-P^) = ^3 ■ (71)
(->00 \ CLK /
The values of [— for different values of n are given in
Table 4.
c) The mass-radius relation. — Eliminating X between (63) and
(69), we have
_5(n+0/(«-.) ^1 . (72)
We shall denote by the quantity
(tL,- <«)
We can re-write (72) as
K = , (74)
where Nn stands for the numerical coefficient
I
n + 1
.oW""’ j
The coefficients Nn are tabulated in Table 4. For the convective
polytropic case, equation (74) is used to evaluate the polytropic
temperature for a configuration known to be a poly trope of a speci-
fied index n of given mass, M, and radius, R.
d) The ratio of the mean to the central density. — Let p(^) denote
the mean density of matter interior to r = a^. Then,
m
M{^)
STa3j3 »
(76)
POLYTROPIC AND ISOTHERMAL GAS SPHERES
99
or, by (67),
(77)
or, since X is the central density, we have
Pc = X =
I JL‘
3
(78)
Relation (78) shows that for a polytrope of a given index, n, the
central density is a delinite multiple of the mean density. The fac-
tor by which we have to multiply the mean density to obtain the
central density for dilTerent values of n are given in Table 4, col-
umn 4. This column, incidentally, brings out a very important fea-
ture of the jxdytropes as a class. The comparatively small range of
n (where o n ^ 5) includes a variety of density-distributions, in-
cluding the two limiting cases of the uniform distribution of density
and the infinite concentration of the mass toward the center.
c) The ccnlral pressure. Since dn = 1 at $ = o, we have
r , = . (79)
Substituting (74) and (78) for /\ and X, resi)ectively, we obtain,
after some minor transformations,
P. =
where ll’„ stands for the (juantity
(80)
II'
n
47r(// + 1 )
(8i)
'The values of ll'„ ari‘ givi'u in 'lablt' 4.
We are now in a position to see how a knowledge of the Lane-
Ihnden functions (‘nal)l(*s us to del(‘rmine the C()mi)lete march of
l\ p, etc., in an (‘fiuilibrium configuration of a given mass, Af, and
radius, A, and known to be a |>olytrope of a si)ecified index, n. From
lOO
STUDY OF STELLAR STRUCTURE
our definitions it is dear that ff* and 6"'*'' give the density and the
pressure in the scales in which pc and Pc are regarded as units. For
a given M and R we can calculate the mean density, p, and (78)
enables us to calculate pc. In the same way, equation (80) enables
us to calculate P.. Finally, equation (74) determines K. Equations
(68) and (77) then describe further features of the configuration.
7. The potential energy. — ^Let V be the potential. Then, according
to equation (8), chapter iii,
p dr dr
From P = we easily find that
By (82) and (83), we have
(»+!)-= -V+V., (84)
P
where Vi is the potential at the boundary. By equation (lo), chap-
ter ii{ equation (84) can also be written as
— V = (« + i) “ H — ^ - (85)
Again, by equation (16), chapter ii, the potential energy, Q, is
given by
2 = ^ j'\dM(r) , (86)
or, substituting for V its value (85), we have
-£2 = §(« + i) ^ dM{r) + i , (87)
or, if dV is the volume element,
X * T mi/f^
FdV + -^^. ( 88 )
POLYTROPIC AND ISOTHERMAL GAS SPHERES 103
other, by means of the homologous transformation 0(Q — » A^d{A^.
{0(^)} is then said to define a “homologous family’’ of solutions.
Thus, { 0 n(S)} defines a homologous family of solutions which are all
finite at the origin and which further have dd/d^ = o at the origin.
We may say that the one boundary condition dd/d^ = o at $ = o
defines the family of solutions Wc shall see presently (§ 9)
that the condition that B shall be finite at J = o already defines the
family We shall refer to solutions belonging to the family
{ 6 n{^)} as “£-solutions.”
Now, the Lanc-Emden equation is a differential equation of the
second order, and consequently the general solution must be charac-
terized by two integration constants. But, as we have seen above,
one of the constants must be “trivial” in the sense that it merely
defines the scale-factor A. It is clear, then, that we should be able
to reduce the equation to one of the first order.
'Jlius the variables used in § 3(d) already enable us to reduce the
Lane-Emden equation to one of the first order.
'J'he variables chosen arc (Eq. [23] with ^ = i, and Eq. [26]):
J ; z = ^^0 ; d) = — - — . (loi)
71—1
z then satisfies the dilTerential equation (Eq. [28] with A == i)
~ + (2<L - i) ~ + w(w - i)z + s’* = o . (102)
We now introduce the new variable, y, defined by
(103)
(104;
Equation (102) then becomes
y ^ + (2w — i)y + (iCw — i)s + z** = o , (105
(IZ
y =
dz
dt
Then,
d^z
(ly
di
dy dz
dz dt
dy
104
STUDY OF STELLAR STRUCTURE
which is an equation of the first order. The reason for this reduc-
tion of the order of the equation is that the functions y and z, de-
fined by
z = ir9 ,
(106)
and by
dz „ dz -a 1 ,.
^ dt hi 1 ai) ’
(107)
or
(108)
are both invariant to homologous transformations. To show this, let
d(^) be a solution of the Lane-Emden equation and let z(|) be the
corresponding function defined as in (106). Let further, d*(^) be
obtained by applying a homologous transformation to 6Q), so that
0*® = A^0(A ^) . (109)
Let z* (^) be the corresponding function defined as in (106) . Consider
the corresponding points 5 and ^/A on the solution-curves 9 and 9*,
respectively. Then
z*WA) = (i/A)^e*{^/A) , (no)
or by (109)
z*WA) = {^/A)-A«9iO = mi) = 2(0 • ( hi )
In words, there is a one-to-one correspondence and an equality be-
tween the set of values which z takes along a given solution and the
set of values which it takes along a solution homologous to the origi-
nal one. This proves that z is a homology-invariant function. To
show that y is also homology invariant, it is sufficient to show that
m^d9/d^ is homology invariant. As before, if we consider the cor-
responding points ^ and A along 9 and 9*, respectively, we have
e>m) - A^m : (f ),.,,^ .
•'‘"'(I),-.'
Hence
POLYTROPIC AND ISOTHERMAL GAS SPHERES 105
or
(5/yl)ai+r
(114)
which proves that is homology invariant. Hence, as y and
s are both homology-invariant functions, we have a first-order dif-
ferential equation between them.
It is possible, of course, to construct other homology-invariant
functions, and we can therefore derive an arbitrarily large number
of first-order differential equations all equivalent to the Lane-Emden
equation. As another example of such a first-order equation, we shall
consider the following two functions, u and zj, defined as
u =
(ns)
where we have used 0' to denote dd/d^.
We can show that u and v are homology invariant by the same
kind of reasoning that we adopted to prove the homology invariance
of y and z. I'he first-order equation between u and v can be ob-
tained as follows:
We have
7/. j 0 "o' *
(ti6 )
Since, according to the Lane-Emden equation,
e" = -jd',
we can re-write (ti 6) as
or,
i du
u d^
1
3 + » Y +
0 ' ’
I du
n tl^
= I (3 - Wf - «) •
(tl?)
(ti8)
(119)
I I ^
V di J 0 **“ 0' ’
(120)
Now we have
STUDY OF STELLAR STRUCTURE
io6
or, again by (117), we find
I I /• I I N / ^
a ii =
From (119) and (121) we have
u dv + t; — I , .
_ — (122)
V du u + nv — $
which is the required first-order differential equation. We shall re-
turn to the foregoing equation in § 21.
We pass on now to a general discussion of the Lane-Emden equa-
tion. The fundamental problem is the following: If we prescribe the
value do and its derivative Bo at a given point fo, then the Lane-Em-
den equation specifies uniquely a solution-curve passing through the
point (So, Bo) and in the given direction prescribed by Bq, The prob-
lem is: What is the nature of such a solution for all the points (fo, Bo)
in the (S, B) plane and for all possible starting-slopes? In other words,
what is the arrangement of the solutions of the Lane-Emden equa-
tion? We shall only be concerned with values oi n > i. For n = 1
the solution can be given explicitly, while for w < i there seem to
be formal difficulties of a far deeper character than those encoun-
tered for n> I. The solution for n = o, however, is explicitly
knoWn.
Q, T he 'E-soluHons. — We shall prove that solutions of the Lane-Em-
den equation which are finite at the origin necessarily have dB/dS = o
at S = o, and that, consequently, the homologous family in-
cludes all the solutions which are finite at the origin.
Consider the Lane-Emden equation in the form (Eqs. [13]
and [14])
d^ jn-i
(x = ^0- (123)
Solutions which are finite at the origin in the ($, B) plane correspond
to solutions passing through the origin in the (x, S) plane. We have
polytropic and isothermal gas spheres 107
Hence,
(‘M) = lim
\d^Ji=o {=0
dx
(12 s)
Since we are considering solutions passing through x O) ^ °>
we can write
^ \ 7 $ )i^o 2 \de /£“0
di \di)(=o V'/?V£--°
By (125), (126), and (127) wchave
/de\ __ T / <H\
\</J/{=o ” 2 \(/S" /£ 0 ’
+ •
+ •
If)
(126)
(127)
(128)
(129)
or, according to (i2;0)
Cl).
since = 0 is finite at the origin, and, further, x = o at $ - o
for the solutions considered. 'I'his proves the theorem.
10 The (y, We shall discuss the solution-curves in the
( V, "‘plane Thchinctions y and =, as we have shown, are homology-
invariant functions, and consequently each so ution
(v '■) plane corresponds to a complete homologous family of solu-
(v -) plans which airrcaiximls lo the K-sukilions which arsincUultJ
„ 'the horn,. logims lamily I ».( £) 1 . We ahall call the eetve which » -
L,»in,ls to the family l«.(l)l the “Km.leii-curve, or the A-
curve,” and denote it by yj.;(s)-
'Ip repeat, our eefuations are
where
-1- (.!<o - Oy-b tl'fw — i):
^ dz
= ^“0,
- o
(131)
108
STUDY OF STELLAR STRUCTURE
and
Further,
y = - ^(ud -1- id') = - - wz .
dz
(132)
(133)
We notice that, according to these transformations, different direc-
tions through a fixed point in the (f , 9) plane correspond to different
points on a definite line parallel to the y-axis.
From the foregoing equations we can derive the following formu-
lae, which We shall need.
From (132) we have
If we denote dy/dz by y', we have, according to (130),
../ _ (2w - i)y + - i)z + . .
^ "" y > (13s)
or, substituting for z and y according to (13^) and (132), we have
/ _ (20) — 1)^6' H- cj^d —
^ + ^0 • (136)
From (136), solving for 0', we have
— a)*0 — (bdV
~ • (137)
We see that the origin, y = z = o (which we shall call O.), is
a singular point of the equation (130), since, when y = o and
2 = 0, dy/dz is indeterminate. In the same way, if » > 3, <5 < i,
the solution has another singular point :
y = o , z = z, = [w(i - w)]'/(>>-i) (ijg)
We shall call this singular point O,. The existence of this second
singular point Oj corresponds to the existence of the proper singular
solutions that exist for w < i; for if w < i, then, as we have al-
ready seen (Eq. [22]),
(139)
= [w(i —
POLYTROPIC AND ISOTHERMAL GAS SPHERES 109
satisfies the Lane-Emden equation. Equation (139) is equivalent to
z, = [m(i — «)]"■'* (140)
in the (y, z) plane, which is identical with (138).
Finally, we have the following correspondence between the (y, z)
and the ($, 6 ) planes. From (135) we see that along the y-axis
(s = o)
y' = -(ato — i) . (141)
But from (136), y' takes the value —(aw — i) for 6 = o. Hence,
the y-axis corresponds to the ^-axis.
II. Behavior near the singular point y = o, z = o. — It is clear that
the £-curve which is characterized by = o at ^ = o in the (g, B)
plane must pass through the origin (sec Eqs. [131] and [132]); and
we have, according to (136),
( y ' ldo , = -W = • (142)
Tfr — I
HcMice, the /i-curve touches the line y + cos = o at the origin. On
the other hand, we can show that there cannot be two solution-
curves which are both tangential to y + ws = o at the origin. To
prove this, suppose y and y* are two diderent solutions such that
y ~ Jis ; y* o) ^ (l43)
We may, without loss of generality, assume that y < y* near the
origin. Then we should have
A = y* - y > o ; lim ^ = o . (144)
Scoo «
From the dilTerential equation (130) we have
^ = [w(w - i)z + z"] ^ . (I4S)
Since yy* ^ co-'s^, we have from the foregoing that
A = o
w(&)— i)z’ + z"'*'*
37 *
]■
(146)
no
STUDY OF STELLAR STRUCTURE
or
( d log A\ _ w(w - i)
'SUiog.j" »■ ■
(147)
But
lim lim
s=0 d log Z s«o log Z
A— 0 A=o
(148)
Combining (147) and (148), we have
log A I
lim 2 — = I — _
s=o log 2 CO ’
A=io
(149)
or
log -
lim= — ^ = - 4 ,
a=o log 2 CO ’
A»o
(150)
which leads to a contradiction since, according to (144), wc should
have a non-negative limit for the left-hand side of (150). This proves
that the E-curve is the only solution-curve vohich is tangential to
y + (iz = o ai the origin.
The line 3; + cos = o has further significance. By (134), along
this line 0 ' = o and 0 ' < o only above the line y + ws = o. Wc
shall refer to the direction y + = o as the ^‘F-direction.”
Now the origin Oi, as we have already seen, is a singular point of
the differential equation; we shall now investigate whether the dif-
ferential equation characterizes directions other than the F-direc-
tion along which solutions can start at the origin.
From (13s)
W)v.z-o = -lim [(2W - i) -f w(w - i) - +-| , (151)
3-,a=oL y y\
which (since w > i) is easily seen to be equivalent to
(/).. *=0 = - (2« - 1) - , (152)
\y Jy, 2=0
or t=o satisfies the equation
y'‘ + (2w — i)y' + «(co - i) = o . (153)
POLYTROPIC AND ISOTHERMAL GAS SPHERES in
Hence,
y = -w or y = -(w - i) . (154)
Thus there is a second direction defined by y = -- (co — i)s to
which solutions can be tangential at the origin. We shall refer to
this second direction, y + (w — 1)3 = o, as the “AT-direction.”
If we substitute y' = — (ci — i) in (137), we have, correspond-
ingly,
Since 3 = 3 = 0 implies ^"0 = o or since o) = 2/(n — i), s = o
implies = o. Hence, according to (155),
If 3 = o corresi)<)n(ls to S = o, then, from (156) it follows that if 0
remains ])ositive, 0' — > — 00 as >0. On the other hand, from
it follows that 0' ~> — c», >0, implies that x(= is fi-
nite at the origin or 6 —> 00 as $ o. But this is true only if s — o
when ap])roaehe(l along the A'-direction corresponds to / — > +00 or
>0. We shall see, however (§§ 19, 20), that under certain cir-
cumstances in ^ 5) the origin 0 , approached along the A'-direction
corresponds to / --> — 00 or ^ > 00.
12. The case ia ^ \ , We shall now consider in greater detail the
behavior of the solutions in the immediate neighborhood of the
origin.
If we are in a sulficienlly close neighborhood to the origin and if,
further, ci i (i.e., n ^ 3 but n > i), we can write equation
(130) as
y + (2(i - \)y + w(w - 1)3 = o , (157)
(iz
or, since y = dz/dl, we can re-write the foregoing as
-f- (2W - 1) 'jy + w(J> - l)s = O . (158)
II2
STUDY OF STELLAR STRUCTURE
The general solution of (158) is seen to be
z = , (159)
where A and B are two integration constants. Since y = dz/di, we
have
y = — — (w — , (160)
From (159) and (160), we obtain
3^ + 032 ; = , (161)
3; + (w — 1)2; = — . (162)
From the foregoing, it follows that
[3^ + (w - 1)2;]^* = C[3; + w2;]“ , (163)
where C is a constant.
We shall choose the X and the Y directions as defined, respective-
ly, by y + (co — 1)2! = o and y + cis = o, as defining a new frame
of oblique system of axes. Let '&x and be the angles which the
X and the Y directions make with the 2;-axis. Then
tan = — (w —• i) ; tan = — w . (164)
Let X and Y be the co-ordinates of a point with respect to the new
system of axes. Then we have
2; == X cos + F cos , (165)
3; = X sin + Y sin •dy . (166)
From the foregoing, we find
_ y cos — z sin dy _
— ~~r’. 7"o o ^
cos dy
sin (dx “ ^y) sin {^x ~ ^y)
(y
z tan d^y) , (167)
or by (164)
cos
sin {dx —
X =
(y + wz) .
(168)
POLYTROPIC AND ISOTHERMAL GAS SPHERES 113
Similarly,
- - .in +
Hence, the solution (163) can be written as
7“-^ = CX- , (170)
or
Y = . (171)
Since w = 2l(n — 1), equation (171) can also be written as
Y = CX^n^-n) ^ (j^2)
From (172) it follows that we have to distinguish again between the
cases (h > i and w < i, i.e., n < 3 and w > 3; the case w = 3 re-
quires special treatmcMit.
13. The case ci ^ i, n ^ 3. "rhe first part of the discussion for
w > i is valid also for the case w = i.
dlie (lilTerential equation can be written as
^ = -(2<i - i)v - - 1)2: - , (173)
2 =
(h
7 fl
= y ,
(174)
with
</v X . V w(w — 1)3 + 3 ”
Y- = — (2aj — l)
(I z y
S = ^"0 = cr^'O ; y = _^<ii ■ ^
(i 7 S)
(176)
Since we need to consider only 0 ^ o, 3^0, we shall therefore
restrict ourselves to a discussion of the solution-curves in the half-
plane in which z is positive. Further, if to ^ i, then, in the half-
plane considered there is only one singular point, namely, O,.
It is this last circumstanc(‘ which makes the discussion relatively
simple, for the solution-curves of (175) must form a one-parametric
family of curves at all points in the half-plane considered, except
at the singular point 0|.
STUDY OF STELLAR STRUCTURE
1 14
From (173) and (174) it follows that along the 2-axis, 2 = o (or
that the solution-curves, if they intersect the 2-axis, must do so ver-
tically) and that the locus of points at which y vanishes is given by
(2(io — 1)3^ = —o){co — 1)2 — 2” , (177)
a curve which (since « ^ i) lies entirely in the lower quadrant.
In the three regions marked /, II, and III we have the signs of
y and 2 as shown in Figure 6.
Finally, the locus of the inflcx is obtained by differentiating the
differential equation (130) and setting d^yjdz^ = o. We obtain in
this way,
(^) ^ - l) + = o • (178)
POLYTROPIC AND ISOTHERMAL GAS SPHERES 115
Eliminating iylis between (175) and (178), we obtain, after some
elementary transformations, for the locus of the inflex
y
1 — i)z -j- Z"
2 w(a) — i) + »z’‘~
-[(2w — i) ± Vi — 4MZ’* .
(179)
As s o, we obtain from the foregoing for <0 i
Hence,
y = — i) + i] (w 7^ i) . (180)
y — — wz or y = — (w — i)z (w i) . (181)
or, at the origin the locus of the inflex (for w i) touches the X
and the Y directions as defined in § ii.
We shall now prove the following lemmas, due to E. Hopf.
Lkmma 1. Any solution-curve y{t), z{t) starting at a point on
the positive y-axis falls monotonically with y decreasing and z in-
creasing and intersects the z-axis vertically at a finite point, after
which the solution continues to fall monotonically with both y and z
decreasing until it intersects horizontally, at a finite point, the curve
V = o; after this the solution rises monotonically with y increasing
and z decri'asing and cither reaches a point on the negative y-axis
for a linite value of I or tends toward the origin as t > 00 ,
'I'liis is intuitively obvious from Figure 6. To prove it, let z{Q = o
and v(/.,) = y„ > o. As long as y > o, z (0 increases, while yit) de-
creases.
Now, since w ^ 1 , vre have, according to (173))
y < - z” .
(182)
Let I, > L be sulliciently near Then y(b) < yo and z(/i) ^
s. > o. By ( 182), y < - z”, for i > L, as z must increase m the posi-
live ((u ad rant. Hcmicc,
y ^ y(/.) - z’;{i - /.) < yo - - ^0 • ^^^ 3 )
Therefore, the curve must cross the z-axis for ^
After crossing the z-axis vertically it is clear from Figure 6 tha ,
so long as we are in region //, y decreases and z also
ly the solution-curve cannot avoid the curve y = o. After crossing
ii6
STUDY OF STELLAR STRUCTURE
this curve horizontally, it is dear that y has to increase while z con-
tinues to decrease. There are two possibilities; either the curve re-
mains in the J/J-region for aU large values of t or it remains there for
only a finite interval in t. In either case the curve must tend to a
limit point, z* ^ o, y* ^ o. In the first case the curve must nec-
essarily approach the singular point, Oi. In the second case z* = o
and y* < o.
Lemma 2. — ^Every solution-curve must be of the form described
in Lemma i.
We have to show that any solution-curve must for t decreasing
reach a point on the positive y-axis.
We shall consider the most unfavorable case, namely, a solution
starting in the ///-region. In this region y > o, and hence, by equa-
tion (177),
y <
— l)g + 2^
(20) — l)
(184)
On the other hand, if y is negative, we have from (175) that
^ - (2w - i) , (18s)
or
y ^ yi - (2« - i)(z - Zi) , (186)
where yi and Zi define the initial point. For large z, (184) and (186)
are contradictory, and consequently z cannot tend to infinity for
the solution-curve in region III. Hence, the solution-curve must en-
ter the //-region for some finite z. In this region, z continues to
increase (as t decreases), while y begins to increase. If y^ < y, then
so long as y is negative, \y^\ > |y| . By (175),
dy ^ / -
i) +
— i)z + 2”
. Iy>l
(187)
where we can choose for y* the value of y at the intersection of the
solution-curve and the curve y = o. From (187) we derive
— i)z^ H
n 1
M
y > — (2w — i)z -b
+ C, (188)
POLYTROPIC AND ISOTHERMAL GAS SPHERES 117
where C is a constant. From (188) it follows that y must become
zero for some finite z and hence must intersect the z-axis (vertically).
After crossing the z-axis, y increases and z decreases (for t continuing
to decrease); since iyjdz is bounded, it is clear that the solution
can be continued to a point on the positive y-axis. This proves our
second lemma.
Consider now the solution y(z; y„), which intersects the negative
y-axis at y„ < o. Lemma 2 has described the character of such a
solution. Then, since the y(z) curves form a one-parametric family
(except at the singular point), it is clear that for suflBciently small z
(in the lower quadrant) we should have
y(3; y») < y(z; yo) (189)
if
yo < y« • (190)
Let us now consider the limit function
lim y(z;y„). (191)
ya-;> — o
Wc shall show that this is the ii-curve, y/o(c).
'lo show this we compare y with another function w which satis-
fies the differential equation
^ -(20) - 1) - [w(w - i) + (192)
where e(< J) is a constant. Equation (192) can also be written as
(ilV
XV h (2cj — i)w -f- [w(aj — i) ■+■ ejs = o .
tiz
'I'he foref];<)in|,5 equation is of exactly the same form as equation
(157), which wc have already considered. Analogous to the solu-
tion (163) of (157), we now have
(
\ Wo
XV + qiZ
Wo
(194)
ii8 STUDY OF STELLAR STRUCTURE
where w ~ nontax z — o and — gi and —q, are the roots of the equa-
tion
q’ -f (2w — i)^ -f [w(w — i) - 1 - e] = 0 .
(19s)
We write
1 + 1 V I — 46 ,
(196)
ga = W — ^ — jV I — 4€ .
(197)
The quantities qt and are real if e < We have, accordingly, as-
sumed e < i- We shall consider the special solution of (193) which
is obtained from (194) by making Wb-^ —0. We have
w = z .
(198)
Since
^ = - (2W - l) - [w(w - l) -1- Z"-'] ^ ,
(199)
we have
-(2«- 1)
(200)
if
gW-I 3; < 0 .
(20t)
Subtracting (200) from (192), we have
_ (jy _ y) > q. ^ [z < («)'/<" y < o] . (202)
We write the foregoing in the form
^ > [w(i - (i) -f 6]2 ^ [z < y < o] , (203)
where
A = w - y = -q,z - y{z; y„) (y„ < o) , (204)
according to our choice of (198) as the appropriate solution of ( 192).
From (204) it follows that
= — y„ > o .
(-’05)
POLYTROPIC AND ISOTHERMAL GAS SPHERES 119
From (203) and (205) we conclude that, under the circumstances
specified, A is positive and increases. Hence,
— qiZ — y{z; yo) >0 [3/0 < o; 2 < . (206)
Now, as € < we have
Vi — 4€ > I — 46 . (207)
By (196), therefore,
> w — 26 > o [w > I ; € < j] . (208)
Hence, by (206) and (208),
yCs; y.) < -qiz < — (w — 26)2 . (209)
Equation (209) is valid so long as y < o, s < e < and
hence so long as 2 < € < J without the restriction y < o.
'Fhe inequality y,, < o is, of course, essential.
lu)r a given i>ositive value of z < we can choose e to be
(2)" '. Hence, by (20()) we have
or
y{z; y.) < -(w - 22" 0- [o < s < (})'*“*] , (210)
y{z; y.) < —cic + , (211)
which is an ine(jiialily due to E. Jlopf. Consider the limit function
yi.]{z) as Vo — > — o (we anticipate by our notation that the limit func-
tion is, in fact, the /veurve).
I'Vom (211) it now follows that
y/':(s) —ws + 23'* .
We will now show that
(212)
yi<(^) ^ -ws (s > o) . (213)
'Vo |)r()ve (213) we proceed as follows: From (iO()) we have in par-
ticular
<JyK
~(h
> — (’oj — 1 ) — a)(co
I)
v/':
(y.; < o) . (214)
120
STUDY OF STELLAR STRUCTURE
In equation (192) choose e = o (we are now considering the equa-
tion [157]). We now have
^ (yE - w) > — — {yE - w ) . (215)
dz ' ymw
Choose for w a solution such that, for z = o, ?£; = < o. Since
= o at z = o, {yE — w) = —Wt^ > o at z = o. Equation (215)
now shows that, so long as y^ < o and w <0, {yE — is posi-
tive and increasing. Now let — o, in which case we obtain
the solution (198) with € = o, i.e., the solution w —cbz. Hence,
we obtain
yE{z) ^ — wz (yE < o) . (216)
From our lemma it now follows that the inequality (216) is always
true.
From (212) and (213) it now follows that, as z — > o, yE must be-
come tangential to the line y + oiz = o. We have already seen in
§ 1 1 that the £-curve is tangential to the line y + oiz = o at the
origin and that there can only be one solution with this property.
Hence, the function defined as the limit y(z; yo), yo — > — o is, in
fact, the £-curve.
We have now shown that the £-curvc passes through the origin
and is tangential to the direction y -f oiz == o at the origin. Draw
the complete £-curvc. Let this curve cut the y-axis at yo{E), It is
clear that a solution starting at a point yo of the positive y-axis
with a value for yo < yo{E) must necessarily remain entirely in the
region bounded by the £-curve and the part of the y-axis o ^ y ^
yo(£), and, according to our Lemma i, must tend to the origin
as ^ > CO (or J o). We shall refer to such solutions as “Af-solu-
tions.” (As we shall see, along an ikf-solution, 0 — ^ 00 as ? — > o,
monotonically.) On the other hand, a solution-curve starting at a
point yo > yo(£) must remain outside the region bounded by the
£-curve and the part of the y-axis o ^ y ^ yo(£), and hence must
reach a point z = o, yt < o of the negative y-axis for a finite value
of t, according to the definition of £-curve and to Lemma i. We
shall refer to the solutions outside the E-curve as the “E-solutions.”
Hence, the whole family of the solution-curves is divided into two
POLYTROPIC AND ISOTHERMAL GAS SPHERES 121
regions by the £-curve, the region of the M-solutions, and the re-
gion of the /^-solutions [cf. Figs. 7 and 8].
14. Tke case <a > i. — So far our discussion is valid for w = i as
well. We shall now exclude the case w = i, » = 3 and consider the
case w > I, re < 3. According to our analysis of § 12, the asymp-
totic form of the solutions near the origin is given by
[y -f- (<i - 1)0]“-' = r[y -t- w0]" ; (217)
or, in the oblique system of co-ordinates defined by the directions
A' and Y, we have (cf. Eq. [172])
or
I' = ,
(218)
C finite .
(219)
From (219) it follows that (if n < 3) all the solutions must touch
the X-axis, or, in other words, the solutions must all be tangential to
the line y + (oi — 1)2; = o at the origin except the one y + w2; = o,
obtained from (217) when C = 00 ; this last case corresponds to the
£-curve. Hence, all solutions other than the / 2 -solution passing
through the origin must touch the .Y-direction at the origin.
From (159) we have for the corresponding behavior of 6 near the
origin:
0 = sf " = = A -f- (/— > +°°) , (220)
or
0 = A + ^ (J o) . (221)
If = 0, we get the solution finite at the origin, and hence a solution
belonging to the homologous family [dJO]- For B 9^ o and posi-
tive we have the behavior near the origin of the Af-solutions which
are seen to tend to 00 monotonically as ^ — > o.
15. The case co == i. 'I'he analysis of § 12 does not apply to the
case (i = I, M = 3, and hence the arguments of the last section can-
not be used for this case. The dilTerential equation for n = 3 is
POLYTROPIC AND ISOTHERMAL GAS SPHERES 123
and equation (157) no longer gives the behavior of the solutions at
the origin. However, the discussion of the £-curve (with regard to
its existence and uniqueness as developed in § 13) is unaltered. In
particular, we have the result that the £-curve is tangential to
y + z = o at the origin.
Now, according to a theorem” due to Hardy, any rational function^
H(x, y, y'), is ultimately monotonic along a solution y(x) of an alge-
braic differential equation of the form f(x, y, y') s SAx*"y"y'P = o.
If we apply Hardy’s theorem to the ratio Xi/Xj of any two terms
of the differential equation itself, it follows that, since Xi/Xj has
to be ultimately monotonic, we should have one of the relations
A\- Ai
(223)”
Equation (222) is an algebraic differential equation, and Hardy’s
theorem is applicable. We should ultimately have one of the rela-
tions
dy _ _ . dy
dz ^ ^ ’ dz
(224)
The first of the possibilities leads clearly to the /i-curve; the second is
imiK)ssible since the Af-solutions are all above the curve y + s = o.
Hence, the only remaining possibility is that, as t — > 00 , dy/dz -> o.
Hence, according to (222),
or
l.2(/ -- < j
1/2
(225)
{2 2 ())
where c is a constant of integration. Remembering that ^ = e \
we have
0 =
(227)
" I^'or a i)r<)<)f of llardy’.s tlicorein sec* H. Hardy, Orders of Injimly (“Camlirid^c
'rractsin Mat hematic's and Mathematical IMiysics,” NTo. 1 2), pj). 57 -ho, i()24.
In exe(‘])lional cases, more tluin two terms beinjj; of ecjual highest order, wc may
have A';f^ constant A',- instead of A';~ — .V..
124
STUDY OF STELIAR STRUCTURE
where C = e~‘ is a constant. Equation (227) then gives the behavior
of the Jlf -solutions. It should be noticed that these solutions in the
{y, 2) plane touch the a-axis, which is for this case the A'-direction
as well. Hence, the behavior of the solutions near the origin is still
qualitatively the same as for the case w > i ; the proof given in
§ 14 is not, however, valid in the present case. The results proved
in this section are due to Fowler.
16. Fowler's theorem. — ^We can now proceed to derive the funda-
mental theorem (due to Fowler) concerning the arrangement of the
solutions of the Lane-Emden equation in the (f , h) plane :
To a given set of initial values
t = S-fcJo; (I),
there corresponds a definite point Zo ^ o, y„ in the iy, z) plane :
2o = ; yo -h o2„ = . (229)
Let us specify a certain point (?o, So) in the (^, 0 )-planc and let 0 ,',
correspond to all possible starting-slopes at (^o, S„). 'Fhen the cor-
responding point in the (y, 2) plane describes a vertical line through
(zo, o). Let 2 = z„(E) be the point where the £-curve intersects the
2-axis.
From the arrangement of the different types of solutions in thi‘
(y, s) plane already described at the end of § 13 wc can easily derive
the following (see Fig. 8) :
If 2o > Zo(£), then the solution passing through (s„, yo), where Vo
is arbitrary, is necessarily an F-solution.
If 2o = z„(£), then the solution passing through [z„(E), yo = o] is
an E-soIution, while the solution passing through [2„(E), yo > o] is
an F-solution.
If 2„ < z„(£), then for a given 2o < Zo(E) there arc two points
[zo, yli'^F)] and [20, yo“’(F)] which lie on the £-curve. Of these two
points one must be in the positive quadrant and the other must be
in the lower quadrant. Let yi''(£) > o. Then a solution passing
through the point [20, yo’('E) > yo > yo“'(.E)] is clearly an Af -solu-
tion, while solutions passing through [so, yo > yl,''(£)] and [a„.
POLYTROPIC AND ISOTHERMAL GAS SPHERES 125
yo < yo“'(-E)] are F-solutions. Finally, the solution passing through
Now a characteristic of an F-solution is that z - o for two finite
values of t, or the corresponding solution” 0($) is such that it has at
least two zeros and that, further, it must be characterized by having
a maximum in the interval in which it is positive. 'I'lie characteristic
of an M-solution is that the corresponding Oi^ solutions tend mono-
tonically to infinity as ^ -> o. Near the origin it has definite asymp-
totic forms, according as « > i or w = i (see bcis. l,22i] and [227]).
»(£) is only om- of a homologous family which can he ilerivcd from a given solu-
tioii-curvc in the (y^ s) plane.
126
STUDY OF STELLAR STRUCTURE
We can translate the foregoing results in the ($, 6 ) plane as
follows:
a) Starting-points (|o, ^o) are divided into two classes by a criti-
cal curve d = Zo(£)f " (zo(£) being determined by a). This curve
is the envelope of all the solutions belonging to the family {0n(5) } .
b) Any solution 6(J) starting at a point (^o, 6 o) above the criti-
cal curve is an F-solution with two zeros and fa such that f i ^
lo < fa.
c) If the starting-point lies on the critical curve, all solutions are
again F-solutions except the one which is tangential to the critical
curve at the starting point ; the latter is an F-solution belonging to
the family {^nCf)}.
d) If the point (fo, ^o) lies below the critical curve, there exist two
starting-slopes d'o = and di = and d'o^^\E) < e!}'\E)
< o. All solutions corresponding to 6 'o > 6 'J'^\E) or do < are
F-solutions. Slopes between and do^^\E), i.e., < d'o <
correspond to JIf-solutions which become infinite as f — ♦ o.
For do = do'''^{E) or d'o = d'J'^\E) we have F-solutions.
e) The asymptotic forms of the Jlf -solutions are
(w > i) ;
(w = i) .
(230)
Fowler refers to the circumstance summarized in (a)-(rf) above
by the very convenient statement that “the F-solutions form a grid
for use in analysing the other solutions.”
If we apply the theorem to the special case where the starting-
point (fo, 0o) is on the f-axis, we have
Zo = o ; y„ = . (231)
As a special case of Fowler’s theorem, or directly from Figure 7 or 8,
it is now clear that if y = yo{E), or
(232)
POLYTROPIC AND ISOTHERMAL GAS SPHERES 127
then we have an £-solution. If < ya{E), or
-em, (233)
we have an Af -solution; finally, if
= (234)
so
we have an F-solution. We have an /'"-solution also when yo is nega-
tive.
17. The case w > |, i < n < 5. I'he general discussion of this
case presents somewhat greater dilliculties, since in the positive
half-plane we arc considering there can be two singular points,
depending upon whether co ^ i or .^ < w < i. We therefore do
not expect the discussion of § 13 to be valid in the present, more
general case. In particular, Lemma 1 is not true for < w < i,
but we can prove the following:
Lkmma. Any solution-curve [y(/), z(l)] starting at a point y, < o
on the negative y-axis falls monotonically, y decreasing and s in-
creasing as / decreases, until it cuts horizontally, for a finite s, the
curve y = o; after this the curve monotonically rises, both y and z
increasing (for / continuing to decrease) until it cuts the s-axis at
a regular ])oint (i.e,, for w < i the curve cuts the s-axis at a point
z > z„ = lci( I — wlp''"*). l"or further decrease of /, the curve con-
tinues to rise, with s now decreasing until the curve cuts the y-axis
again at a [wint on the positive y-axis.
We have already i^roved the lemma for w ^ 1, and we shall
therefore restrict ourselves here to the case i > w > J. In this case
there is a singular point on the s-axis (denoted by O,). The foregoing
lemma shows that solutions which cut the negative y-axis (y, < o)
do not ajiproach any of the singular ])oinls in the half-plane con-
sidered, and in all cases for which o) > J these solutions avoid the
singular points.
It should be noticed that the locus of points at which (/y/Jl = o,
s})eciried by (cf. Vj([. [177!)
(2w - i)y = w(.i - w)3 - s" , (23s)
\onget a locus ^.axls at
ee.couu
\eiiuaa.
so loug a-® ^
^ ^ _ (2W
(.23'/')
POLYTROPIC AND ISOTHERMAL GAS SPHERES 129
which we can also write as (for y < o)
dv , . ^ w(i - w)z
5 bl ■
(238)
Further, in repon lit, lyl increases and hence [y,]
mum value of jy] in this region. Hence,
is the mini-
dy , . s w(i - «)s
S IJ.! •
(239)
or
, ^ w(i — oj) 2
3, > y. - (2W - i)Z 2|y,| ^ •
(240)
On the other hand, since we are in region Til, y > o, and we
should have
m(i - m)z - g" _ (24J)
(201 — i)
y <
Eciuations (240) and (241) arc obviously contradictory for large
values of z. Hence, the solution-curve must leave the region con-
sidered for a finite z, and it is clear that it intersects the y = o
curve (horizontally) at a point where z > z,. For further decrease
in I, z continues to increase, while y now begins to increase, jyj de-
creasing. If y jt ^ V, then so long as we are in region IT, |yj| ^ 13 ^ 1 1
and by (2.^6) we have (for sulliciently large z)
> rz"
> - (2W - t) -1
dz
for some positive constant, ( . Ihus,
C’z"''
yd
(242)
y > J) (1) a constant) , (243)
n -t- I
and we conclude that the curve must cross the z-axis. Further, the
curve must intersect the z-axis at a juunt z > z„ since, as we have
already seen, the curve already intersects the curve y = o at a
|)oint whose z-co-ordinate is greater than z„ and in the region IT,
z increases with decreasing /. After crossing the z-axis (vertically),
y increases, while z now begins to decrease; and, since \dy/dz\ is
130
STUDY OF STELLAR STRUCTURE
bounded, the solution can be continued to a point on the positive
y-axis.
Consider, now, the solution y{z; y^, which intersects the y-axis
at a point y^ < o. The lemma has described the character of such
a solution. Then, since the (y, z) curves must form a one-paramet-
ric family (except at the singular points), it is clear that for suf-
ficiently small z we have
y(z; yi) < yi.^; yO < y, < o) . (244)
We can therefore construct the limit function
yE(z) = lim y(z; y,) . (245)
y. -■■> — o
We have already shown that for i ^ i we obtain in this way the
£-curve. We shall see presently that this is generally true. The E-
curve is, of course, tangential to the direction y + <02 = o at the
origin.
Draw the complete £-curve. From the lemma it follows that the
£-curve must cut the y-axis at a point yo(E) > o. Further, it is
also clear from the lemma that the curve surrounds the singular
point (o, 2a). Again, it follows from the lemma that a solution start-
ing with a value yo > yo(E) remains outside the £-curve. This so-
lution intersects the y-axis for two values of t, and hence corre-
sponds to a solution in the (^, 0) plane which has at least two zeros.
We shall refer to these as “F-solutions.’^
On the other hand, if we consider a solution starting at a point
o < yo < yo{E) on the y-axis, then the solution must be entirely
inside the region bounded by the £-curve and the part of the y-axis
o ^ y ^ yo{E). Hence, as > 00 the curve must tend to one or
other of the singular points inside the region described. If oi ^ i,
there is only one singular point, namely, the origin; and, as we have
already seen, the solutions approach the origin. On the other hand,
there are two singular points in the region under consideration for
^ < oi < I. In this case, however, the solutions cannot approach
the origin. This follows from the analysis of § 12. We have shown
POLYTROPIC AND ISOTHERMAL GAS SPHERES 131
that the behavior of the solutions in the immediate neighborhood
of the origin is specified by
F = , (246)
where X and Y are the co-ordinates in the oblique frame of refer-
ence defined by the lines y -t- — i)z = o and y + = o. If
(i < I, we can write
= constant , (247)
where the exponent of X is positive. It follows that, in general,
F = o corresponds to Y = 00, and similarly X = o implies
F = 00 . Hence, the solutions approaching the origin must be such
that they approach it up to a certain minimum distance and then
recede from it. The exception is the £-curvc which corresponds to
C - CO in (246); this solution touches the F-axis at the origin and
hence, being tangential to y + ois = o at y, 2 = o must be the
/i-curve (§ 11). Hence, the solutions in the region bounded by the
/i-curve and o ^ y ^ yo(£) must approach the second singular
point O2 (o, s«) on the s-axis for i > w > |- as ^ 00 . We shall re-
fer to these solutions as the *^M-solutions.’’
'rhe behavior near the singular point for t > w > I will be
examined below, hut it is clear that in the terminology of Fowler
the /i-solutions form a grid for use in analyzing the nature of the
solutions ])assing through a given point in the ($, 6 ) plane, for
w >
18. 7 V/r case i > co > 3 < n < 5. -The behavior of the so-
lutions as they approach the singular point 0 ^ will now be investi-
gated. Let us first examine the i)ossible directions of y' at 0 ^.
Near (), we can write
v' = lini
.V - ► o
As o
-(2cj - l) +
<i(r — «)(s, + As) — (s, + ^s)" I
. (248)
or, since z" ' = «(i — w), we have at (o, s.)
(o, z.) , (249)
132
STUDY OF STELLAR STRUCTURE
or we can re-write the foregoing as
y* + ( 2 cj — i)y + 2(1 — to) = 0 .
(250)
Solving for y', we have
y = |[~(2w — i) ± v'4m“ -f 4(0 — 7] .
(251)
The values of y' at (0, Za) given by (251) are real only if
40)^ + 46) — 7 ^ 0 .
(252)
Let CO = CO* be such that
4i*» -t- 4«* - 7 = 0 .
(253)
The positive root of the foregoing equation is given by
.j, zVz — I
— -= O.9I42I
(254)
w* corresponds to a value of n* where
a, II -b 8V2
— = 3.18767
( 2 SS)
If « < «*, the directions specified by (251) are imaginary. It fol-
lows that the solutions approaching the singular point O2 must spiral
around (o, z,). On the other hand, if i > w > w*, the directions
specified by (251) are real. Let X, and F, denote these two direc-
tions. Then,
y'r. = -§[(2u - i) -b -f 4w - 7] , (256)
y'x. = -M(2w - i) - ^ -f 4 " - 7] • (257)
We shall examine in greater detail the behavior of the solutions at
O2. First consider the case w < «*:
Case i: ^ < u < u* < 1 . — ^Write
z = z,+ z, ; z, = [co(i — &>)]“/“ . (258)
POLYTROPIC AND ISOTHERMAL GAS SPHERES i 33
If z. is sufficiently small, we can write the differential equation (130)
at z = z, + Zi (where z, o), as
y ^ + (2i - i)y - «(i - «)(z. + zO + z? + »zr-z, = o ; (259)
^ dzi
or, remembering that z, = [w(i — ^ we have
y ^ + (2i - i)y + 2(1 - «)z. = O ; (260)
dzi
or, since y = dz/dt = dzjdt, we can re-writc the foregoing as
^ + (2J, - i) ^ + 2(1 - ci)z. = o . (261)
dt
The roots of the equation
+ (2w - i)q + 2(1 - w) = o
(262)
are imaginary (cf. Eq. [250]) when < «*• The roots can be writ-
ten as , _ - . / / \
-K2ii - i) ± - 4“ - • ^^"3)
The solution of (261) can therefore be written as
2^ _ 'U t;(,s [2 V ^7 — 4 w — 4 “' < + 5] , (264)
where A and 6 are integration constants. Remembering that
<0 = 2/(« — i), we verify that
T...'. . 2J) - i = ■ (2(>S)
" ' / \ . » 1;. — T
7 - 4 (L - 4 “' - --■ ( 7 r- i)
We can now write (264) as
I \/>jn‘ - 22« - X , J
,. = ,1. ^('.-0 cos I < + 5
Since y = dz/tll — dz,/dl, we have
_ ,( c-n - 22H - T .1
(266)
\/yn^ — 2 21 1 — 1 I
2 2n
2(w — i)
2(« - l)
l + b
]}■
(267)
134 STUDY OF STELLAR STRUCTURE
We thus see that the singular point (o, z,) is approached spirally
as f , and, since ^ approaching the singular point cor-
responds to $ — > o.
As t increases by 4(« — i)7r/v7^j®-- 22» — i, the representa-
tive point in the (y, z) plane makes a complete revolution, while
the “amplitude” decreases in the ratio
3(5-»)y
I : e . (268)
Again, if f or a certain valu e of t, Zi = o, then if ti increases by
2 {n— i)'3r/v'7M* — 22» — I, Zi will again be zero ; this means that,
as f 00 , 2, = o for I asymptotically decreasing geometrically in
the ratio
— i)t
I ; e . (269)
As z, becomes successively zero, the y-co-ordinate asymptotically
decreases geometrically in the ratio
Again, since
0 = (z, -h z.)c“‘ = (z, -1- z.)?-* ,
(270)
(271)
we have, according to (266),
S-w
_|. QJ(»-x) cos
[
2 2n — 1
2{n — i)
log ?
-s]},
(272)
where C is a constant.
From the foregoing it follows that, as ^ o, the solution crosses
the singular solution at points which asymptotically decrease geo-
metrically in the ratio (269), while (as J > o) the solution becomes
asymptotic to the singular solution
e. =
2(» - 3)1’'''*"’’ I
(273)
9 da
^ o) . (274)
POLYTROPIC AND ISOTHERMAL GAS SPHERES 135
Case 2: I > w ^ < 5 *. — In this case the roots of the quadratic
equation (262) are real, and we can write for the solution
Si = Ac «»* + , (275)
where ~gi and — are the roots of the equation (262);
<7i = i'[(2w — i) + + 4w — 7] , (276)
q2 = M(2w - i) - -j- 4^ _ 7] . (277)
From (27s) we have
3; = — q,Ber'f^^ . (278)
From (27s) and (278) we derive
(>' + qxz,y^ = C{y + giSi)'/« , (279)
where C is a constant.
If we choose the directions yl*,, y'r, (cf. Eqs. [256] and [257]) at
(o, s«) to dellnc a new oblique frame of reference and denote by A'l
and K, the co-ordinates of a point with respect to this new frame
of reference, we obtain, as in § 12,
VV = CXV ;
or by (276) and (277) we have
(joi— i)-4~'n / h«>‘' I -IM — 7
Hence,
Hence, all the solutions except one (which corresponds to C = oo
in [281]) touch the A', axis at (o, s„). A closer examination shows
that this happens as / — > oo. From (275), (276), and (277) it fol-
lows that, since
0 = sc“‘ = (s, -h s,)c“‘ , (283)
we have
0 ~ S,C“‘ + .• 1 cll>-V'. 1 <i'H-.|u- 7 l < -p 7 iclli + \/.i<i' l-.1i.-7l ‘ 00) ; (284)
(280)
(281)
(282)
136 STUDY OF STELLAR STRUCTURE
or, remembering that ^ = e~‘, we have
4 _ + g=. , (285)
^i[l-V' 4 »*+ 4 »- 7 l ^i[l+V' 4 “‘+ 4 M- 7 l
as ? — > o. The “exceptional” case referred to above, which is tan-
gential to the Fi-axis at Oa, corresponds to 5 = o in (285). Equa-
tion (285) gives the behavior of the solutions which tend to infinity
as ^ » o; these are the M-solutions.
It is now clear from the lemma of § 17 that the arrangement of
the solutions is the same as for the case m ^ i, the difference aris-
ing only from the different asymptotic behavior of the ilf -solu-
tions as ? — » o. Instead of (221) and (227) for w > i and w = i,
respectively, we now have (272) and (285) for | < to < «* and
I > w > 03*, respectively. Fowler’s theorem, as stated in § 16,
holds good except for (e), which describes the asymptotic behavior
of the solutions.
19. Case « = I, n == 5. — In this case the (y, z) differential equa-
tion
y ^ - |z -1- zs = o (280)
can be integratedj and we have
y = + D , (287)
where D is an integration constant. As we have already seen in
§ 4, D = o leads to the Schuster-Emden integral, and hence in
the (y, s) plane the £;-curve is given by
yh = — 3Z® • (288)
This represents a closed symmetrical curve tangential to the lines
y ± ^z = o at the origin. The lines y + §2 = o for the case under
consideration (w = J) define the X- and the F-directions. The ori-
gin approached along the line y + = o for increasing t corre-
sponds to $ — > o in the ($, 0 ) plane. At the same time, the origin
POLYTROPIC AND ISOTHERMAL GAS SPHERES 137
approached along the line y = for decreasing / corresponds to
J cx) with 6-^0.
The £-curve intersects the s-axis at
2o(/0 = (289)
If D in (287) is positive, we get curves (symmetrical about the
s-axis) which lie entirely outside the £-curve and which intersect the
y-axis at the points + Vd. These are the F-solutions. If D is nega-
tive, we obtain closed curves inside the £-curve; these are the M-
solutions. As in the previous cases, the region of the F- and of the
M-solutions are separated by the £-curve. Thus, the Schuster-Em-
den solutions in the (f , 6 ) plane also form a grid, and the arrange-
ment of the solutions is of the same nature as in the cases already
discussed. The critical curve in the (^, 9 ) plane above which the
solutions are all of the F-character is deiincd by (cf. Eq. [289])
0 = , (290)
which is the envelope of all the Schuster-Emden integrals (Eq. [59]).
Again, for | the singular point is [o, (J')*'''’], and the corre-
sponding singular solution is
0 . = ,
(291)
which lies below the critical curve (290).
If we consider the M-solutions in somewhat greater detail, we
see that, as >0, the solutions lie below the curve (291), while
with increasing f they cut the singular solution (291) and rise above
it, and when ^ still further increases, they cut the singular solution
again and tend to zero as ^ — > 00. Since the value of z is the same
both when I —> 00 and when / — > — c» ^ we have
lim = lim . (292)
« - * CO 0 ~*0
i - ► CO
The general run of the curves in the (y, s) plane is illustrated in
Figure ii.
Fig. io.— The system of isoclinical curves in Fig. ii.— The system of isoclinical curves in the
the (y, z) plane for » = 4. (The diagram is re- (y, z) plane for w = 5. (The diagram is reproduced
produced from Emden*s Gaskugeln) from Emden*s Gaskugdn.)
POLYTROPIC AND ISOTHERMAL GAS SPHERES 139
20. The case H — The lemma proved in § 17 is valid no
longer. This can be seen in the following way:
The equations are
2w)y + aj(i — co)2; — ,
(293)
y >
(294)
dy
dz
(i
2w) +
2 '*
(29s)
The curve y = o is now given by
(i - 2o))y = -w(i - w)z + z" , (295')
and this curve is initially in the lower negative quadrant; it inter-
sects the 2-axis at (o, z«) and tends to infinity in the upper quad-
rant. The signs of y and z are now as shown in Figure 12.
In contrast to the case i < ^, we can now prove the following
lemma:
Lemma.” -Any solution-curve [y(l), zil)] starting at a point > o
on the positive y-axis, with increasing /, monotonically rises (both
z and y increasing) until it intersects, for a finite /, the curve y = o.
After crossing this horizontally, the curve monotonically descends
with y decreasing but z still continuing to increase, until it inter-
sects the z-axis at a finite iK>int 2 > z«. After crossing the s-axis
(vertically), it continues to descend, with both y and s decreasing,
until it finally intersects the negative y-axis at a finite point.
Let the initial starting-point be (y„ > 0,0). In the region / both
y and 2 increase with increasing /, so that the minimum value of y
in this region is y„ itself. Hence, from (295) we have
dv ^ , a)(l — d))2 /
■ < (i - 2a;) + , (296)
iiz yo
or
y < + (i - 2j))z + -—-7“ 2“ • (297)
2yo
140 STUDY OF STELLAR STRUCTURE
But so long as we are in region I,
Fig. 12
Equations (297) and (298) are contradictory for large values of z.
Hence, the solution must leave the region considered and for a finite
t must intersect the locus y - o. This point of intersection, (y„ z,),
must have its z-co-ordinate Zj greater than z,.
POLYTROPIC AND ISOTHERMAL GAS SPHERES 141
In the region J/, as ^ continues to increase, y decreases while %
continues to increase. From (295) we have
= (i - 2u) jy? + y(i - z«+‘ + c , (299)
where C is a constant. Since y decreases in this region, the maxi-
mum value of y is 3;,, and hence
< (i — 2w)yiS + — w)2^ — - + C , (300)
which shows that must become negative for a sufficiently large
value of s, i.c., the solution curve cannot avoid the z-axis. Hence,
the solution curve must leave the region considered and cross over
into region TIL In this region y and s both decrease with increas-
ing t; since |yl and ls| are bounded and finite in the region con-
sidered (///), the solution can be continued to intersect the y-
axis at a finite nonzero point on the negative y-axis. This proves
the lemma.
'I'he foregoing lemma is of im])()rtance because it shows that solu-
tions starling on the jmsilive y-axis avoid both singular points, just
as the solutions starting on the negative y-axis for i > oi > -J-
avoid the two singular points.
Now since the (y, z) curves form a one-parametric family (ex-
cept at the singular points) for sufficiently small values of s, we
should have
y(s; y..) < y(s; y„) (o < y„ < y„) . (301)
We now construct the limit function
y/;(z) = lim y{z; y„) . (302)
.Vo 4o
Our discussion of § 12 has shown that for w 7^ i, we have
= constant , (303)
where -V and Y are the co-ordinates of the point with respect to
the frame of reference delined by y = (i — aj)z and y = — wz (the
142
STUDY OF STELLAR STRUCTURE
X and the F directions, respectively). Equation (303) shows that
in our case (w < 5), there can only be (exactly) two solution-
curves passing through the origin, one each in the directions X
and F.
It is clear that the solution yoCz), as defined in (302), which passes
through the origin, must be tangential to the X-direction or
[ynCz)]!/. *-o = I - w . (304)
We sha.11 call this the “D-solution,” and the corresponding curve
yoiz) the “D-curve.”
Draw the complete D-curve and let this cut the negative y-axis
at yo{D). An examination shows that the Z)-curve which passes
through the origin in the direction y = (i — <j>)z must do this as
t—* — 03 (this arises from the circumstance that, if we trace the
solution from yo(Z)) “backward” for decreasing t, it can tend
toward the singular point Or only as i — > — ■» [or » -f- »]). In
other words, the origin approached along the D-curve in the y =
(i — «)z direction as y, z — » o corresponds to approaching the
“boundary” in the (§, 9) plane. From equation (159) we find that,
as I — > 00 , we should have
2=5e(i-“)' (f_>_oo). (305)
From this it follows that along the solutions in the {9, plane which
correspond to the D-curve in the (y, z) plane, we should have asymp-
totically,
(f-^oo). (306)
Finally, any D-solution has a zero for a finite value of
From the lemma it follows that the solutions starting on the nega-
tive y-axis at yo < yo(D) must lie entirely outside the Z)-curve;
since these solutions intersect the y-axis twice, they correspond in
the (9, 5 ) plane to 9 having two zeros for finite values of J. In other
words, outside the Z)-curve we have F-solutions.
Now consider a solution which starts on the negative y-axis
for o > yo > yo(D). Such a solution must remain entirely in
the region bounded by the Z?-curve and the part of the y-axis
POLYTROPIC AND ISOTHERMAL GAS SPHERES 143
yo{D). We shall call this region the “O-region.” As we
continue the solutions in the 0 -region from the negative y-axis for
decreasing t, it is clear that, as i ^ these solutions must ap-
proach one or the other of the two singular points in the 0 -region.
They cannot, however, approach the origin; the Z)-curve is the only
one which docs this (this follows from Eq. [303I). Hence, the solu-
tions must approach the second singular point (o, 2«) as i “ •
The discussion in § 18 of the possible directions at (o, z.) now
applies; we conclude that, since m < 5 < w*, the solutions in the
0 -region must spiral around the singular point (o, z,). The discus-
sion of the behavior of the spiraling at (o, z.) runs parallel to the dis-
cussion of the spiraling for the case w* > w > 5; the important dif-
ference, however, is that in the previous case (w* > w > 5) the
singular point is approached as i -h ® , while in our present case
(w < the singular point is approached for - ” ■ In other
words, the solutions in the 0 -region approach the singular solution
0 , = [i(i - , (307)
oscillating as .
More explicitly, the formulae (266) and (267) are now valid as
they stand, but the interpretation is dilTerent: while then » ,
now / — » — CO. In particular, we have the solution (cf. Eq. [272])
0
2(n - ,0 |«
(« — O"
I ^ - o)S
t l»-l)
— 2211 — 1
2 (« - 1)
log ^ — 5
1 (308)
where C and 5 are constants.
From the foregoing, it follows that, as ^ ^ , the solution
crosses the singular solution 6 , at points which asymptotically in-
crease in the ratio
2(h— i )y
J. . ^a/7«'-22»-1 ^ (309)
As ^ ^ 00 , the difference between 6 and also tends to zero.
144
STUDY OF STELLAR STRUCTURE
Finally, if we make —o, we obtain the solution passing
through the origin tangential to y + = o at Oj. This is the E-
curve. This JS-curve also spirals around the singular point O2 as
^ CX3 . In the (5, B) plane this means that a solution belonging to
the family {Sn(^)} asymptotically approaches the singular solution
Bs oscillating as ^ . Further, the points of intersection of any
£-solution and the singular solution asymptotically increase geo-
metrically in the ratio (309). The £-solution differs from the other
solutions in the 0-region in that a solution belonging to the family
{0n(^)} has no zero for any finite value of while in general (i.e.,
yo{D) < yo < o) the 0-solution has a zero. We have, incidentally,
shown that configurations with w > 5 all extend to infinity. (For
^ = 5 we found that the mass of the configuration was finite; but
for n> s, one can easily show from [308] that — > » as
5 — > 00 , and that therefore the mass is also infinite.)
It is now clear that the D-solution separates the region of the
0-solutions and the region of the F-solutions, and hence the Z)-solu-
tions in the ($, B) plane form a grid for use in analyzing the other
solutions, just as the £-solutions formed a grid with the necessary
properties for cS ^ The arrangement of the solutions can be
stated in the following way, which is very similar to Fowler’s theo-
rem:
d) Starting-points B^ are divided into two classes by a criti-
cal curve B = where ZoiP) is the value of z for which the
O-curve in the (y, z) plane intersects the z-axis. The curve
B = 2;„(0)r"
(310)
is the envelope of all the /^-solutions. A D-solution has a zero for
a finite $ and, after attaining a maximum, tends to zero monotoni-
cally as ^ o) ; and the asymptotic behavior at ^ = «> is given by
(311)
All the Z)-solutions form a homologous family.
h) Any solution B(^) starting at a point (^o, Bo) above the critical
curve (3 10) is an F-solution with two zeros at finite points.
146
STUDY OF STELLAR STRUCTURE
c) If the starting-point lies on the critical curve, all the solutions
are /^'-solutions, except the one which is tangential at its starting-
point to the curve (310) ; in this case it is a D-solution.
d) If the point (^0, 60) lies below the critical curve (310), two
starting-slopes, do = do^^^D) and di = exist which lead to
D-solutions. Let > 6o^’\D). All solutions corresponding to
60 > and 6'o < d'o^^\D) lead to F-solutions. Slopes between
and lead to 0-solutions, which are described below.
e) Any 0 -solution tends asymptotically to the singular solution
, _ f 2 (m — 3)] I
(312)
as 5 — » 00 . The singular solution is, however, approached in an os-
cillating manner, i.e., an 0-solution intersects the singular solution
again and again, and, as ^ , the points at which an 0-solution
intersects the singular solution increase asymptotically in a definite
ratio.
f) There exists a special class of 0 -solutions — the F-solutions —
which form a homologous family, and which are finite at ^ = o,
and which have, further, d6/d^ = o at $ = o. Any 0 -solution
which is not an F-solution has a zero for some finite
21. Discussion in the (u, v) plane . — We shall now briefly discuss
the arrangement of the solutions in the {u, v) plane. As has been
already defined in § 8, we have
M
e' ’ ^ e '
(.31.'?)
Further, we saw that u and v satisfy the differential equation
(Eq. [122])
udv___u-\-v— 1 . .
V du u nv — V- /
As may be verified easily, the (w, v) variables are related to the
(y, z) variables according to the relations
Z = ^ , (315)
y = . (316)
POLYTROPIC AND ISOTHERMAL GAS SPHERES 147
The (w, v) variables, first introduced by Milne, have the great
advantage that the positive quadrant (w ^ o, v ^ o) contains only
such parts of the (?, 0) solution which are of astrophysical interest,
i.e., points in the positive {uy v) quadrant correspond to 6 ^ o,
0' $ o.
We shall now consider some general properties of the (w, v) plane.
a) The locus of points at which the solution-curves have hori-
zontal tangents is given by
u + V = I . (317)
b) The locus of points at which the solution-curves have vertical
tangents is given by
u + nv ^ 3 . (318)
c) For w < 3 the loci (317) and (318) do not intersect in the posi-
tive («, v) quadrant.
d) For w = 3 the loci arc
u + V ^ 1 y w + 3^ = 3 • (319)
Hence, they both pass through the point (v = 1- w = o) on the
^^-axis.
<0 For n > 3 the two loci intersect at the point (w«, v^), which
is easily verified to be
= = w . (320)
n — 1
This intersection of the two loci, (317) and (318), in the positive
quadrant corres])onds to the existence of the ])ropcr singular solu-
tion for n > 3 ;
2{n — 3) I
(n - 1)^. »
(321)
^(Nir)/(N «)• t32 2;
If we form the variables ii and v as definecl in (313), we readily find
from the foregoing expressions that u and v reduce to and as
defined by (320).
0: = -2
- 3)
{n - j)"''
148
STUDY OF STELLAR STRUCTURE
f) Let us consider the E-solutions at ^ o. It is sufficient to
consider since any other member of the family {0„(?)} will
lead to the same («, v) curve. As S > o, we have (Eq. [62])
»«(?) = I - ^ - • • • • (« ^o) • (323)
From the foregoing, we find
e;(«) ~ I - I r ; «4(f) ~ ^
Hence, as ^ » o,
me = -^ ^ 3^1 - (f-»o). (325)
aE=-^~i? («->o). (326)
We see that the £-curve passes through the point
Me = 3 , VE = o (^ = o) , 327)
for all values of n. At this point the £-curve has a definite slope
determined by
dUE 2W dVE 2fc
Hence,
{p\ . -i. (3=,)
\duEj(=o 3«
g) Let us consider a solution for « < 5 which starts with a defi-
nite slope on the ^axis at ^ = $.• According to Fowler’s theorem,
there is exactly one £-solution through the point. All solutions
with starting-slopes below the £-curve are AT-solutions, and all
those with starting-slopes above the £-curve are F-solutions.
Given the slope at we can easily form a Taylor series at this
point. From the Lanc-Emden equation we have
e" = -e™ — I O'. (330)
POLYTROPIC AND ISOTHERMAL GAS SPHERES 149
Since 0 = 0 at g = ?i, we have
1
11
(331)
Differentiate (330) and set ? = We find
11
1
(332)
or by (331)
(333)
The Taylor scries in the neighborhood of is given by
; (334)
or, using (331) and (333)) we obtain
0(0 = (? - o) 0 {. 1, % "^ 0 ■ ■ ■
■ , (33s)
or
*<£) - f, * + C' £, *) + ("ir)
• { 33 <>)
Let ,
UA.-
(337)
Now the function — ^“+' 0 ' is a homology-invariant function, and,
since the zeros of two members of a homologous family are “corre-
sponding points,” it follows that any homology-invariant function
(and therefore also -^+' 0 ') will have the same value for all the
members of a homologous family at their respective zeros. Since
each homologous family yields only one curve in the («, f) plane,
and since, further, every member of the homologous family will
have the same value for a)„, it is clear that we can choose a)„ to label
the solution-curves in the («, v) plane. In particular, the A-cun^e
will be labeled by the ciuantity «<»„ already introduced in § 6 (cf.
liq- [vsD-
STUDY OF STELLAR STRUCTURE
ISO
From (336) and (337) we can write
Hence, we have
as S From (340) and (341) we finally have
(338)
(339)
(340)
(341)
^ co; * . (342)
In other words, as w — > o, —> ® . In particular, along the £-curve
[uv^]e ^ oO)T^ (w o) . (343)
Now, from Fowler’s theorem we have (cf. Eq. [231], [232], [233],
and [234])
MWn < oWn < pO)n • (344)
The M-curves, therefore, lie inside the region bounded by the
£-curve, the 7;-axis, and the part of the w-axis o ^ m ^ 3.
The F-curves, on the other hand, lie entirely outside the E-curve.
Also, when an F-solution attains its maximum in the {d, plane
at, say, Jo, 0 o (where Jo > o), it is clear from the definitions of the
variables u and v that along the corresponding F-curve, co
and V o. Further, since
uv = , (345)
it follows that the F-curves tend to become like rectangular hyper-
bolas when they asymptotically approach the w-axis.
h) Finally, let us consider the behavior of the (w, v) curves near
the point (w = 3, ^; = o). Write
W = 3 + .
(346)
pqlytropic and isothermal gas spheres 1 51
The (m, v) differential equation (314) now takes the form
3 + M. ^ _ 2 + M. +J> _ (347)
» dui nv + Ml
If Ml and D — ♦ o, wc can write, approximately,
3 = ? — . ( 348 )
V dui nv + M,
A separation of the variables can be effected by the substitution
D = UitV .
(349)
Equation (348) reduces to
3 (fw S "E 3 ;'”?’.
w dill Ui{i + ^
t 3 So)
or 1 j
1 S + 3 ^^^ ^
(351)
inlcgralinf^, wc lind,
(352)
= constant ,
which wc write as
(5//1 -h = constant .
(353)
Returning to our varialiles u- and t>, we have
v'/-’Is(m — 3) + 3'«’1 = eonstanl .
(354)
Hence, near 11
generalized hyj
= 3 .
)erl)olas
= o, the ill, 1') curves resemble portions of
asym])totic to the M-axis and the line
5(m - 3 ) + 3'”' = o •
(355)
We get the A-curves when the constant
the M-curves when it is negative. When
get the li-curve represented near iu =
(355); thus we arrive at our earlier result
in (354) positive, and
the constant is zero, wc
3, V = o) by its tangent
(cf. Eq. [329])-
IS2
STUDY OF STELLAR STRUCTURE
We will now consider a little more carefully the different cases :
I < « < 3, « = 3, 3 ^ ^ «* < » < s, w = s, and n > $■
The principal question to be considered is the nature of the M-
curves.
Case i: I < n < 3. — ^We have already described the E- and the
jP-curves (/, g, and h above). Now, along an Jlf -solution as ^-^o,
Fig. 14. — ^The («, ») curves {n = 1.5)
we have the asymptotic rela-
tion (Eq. [221])
(356)
and correspondingly, as may
be verified,
« = o, I) = I . (357)
Hence, the general nature of
the (m, v) curves are as shown
in Figure 14.
Case ii: n = 3. — According
to (225), along an Jlf-curvc
we have
z-^o). (358)
From (316) it follows that (as ai = i in our present case)
— (uvy^^ = (z^ 7)3 )i /2 __ ^
(359)
or
UV V — 1. = 0 .
(360)
But we should also have (Eq. [315})
z = {uvY^ — » 0 as 1 0 .
(361)
From (360) and (361) it follows that the M-curve
V = i) as ^ 0.
tends to {u = o,
POLYTROPIC AND ISOTHERMAL GAS SPHERES 153
Hence, the arrangement of the curves is qualitatively the
same as in case i above (cf. Fig. 15).
Case Hi: 3 < n < n* =
3.18 — The two loci
u + V = I and u +nv — 2^
intersect at the point
Vs) (given by Eq. [320])
in the positive quadrant.
As may be verified from
our asymptotic formulae
for the M-solutions for this
case (Eq. [285]), the Af-
curves in the (u, 7)) plane
tend to the point (w*, Vs) in
a definite direction.
Case iv: n* < n < 5. —
The AT-curves now spiral
around the point (//«, v^), but the nature of the E- and F-
ciirves are as before (cf.
Fig. 16).
Case v: n = 5. - As in
the (y, s) plane, the equa-
tions of the (w, z^) curves
can be explicitly given.
From (287) we have
I’n;. i().— The (//, ?•) curves (w = 4 ) y = J 4" • ( 3 ^^^)
Further, for this case we have (Ec^s. [313] and [31 6])
z = Civy^^ ; y = (z*z;5)«/4 — \z , (363)
I'll]. 15. — The (?/, v) curves (« = 3)
Substituting (363) in (362), we have
(iiv^yi^ -f — z{uv^y^^ = Js' — !i(uv}^''^ + D , (364)
or
Civ^yi^ — = —\{uvyf^ + D . ( 365 )
IS4 STUDY OF STELLAR STRUCTURE
Dividing throughout by and rearranging the terms, we find
that
« + 3® = 3 + ■ (366)
The Schuster-Emden integrals correspond to D = o; hence for
» = 5 the £-curve is the straight line
M + 311 = 3 . (367)
The arrangement of the (u, v) curves is as shown in Figure 17.
U “>
Fig. 17. — The («, v) curves {n = 5)
Case vi: n > 5. — Now the general nature of the zO curves is
completely different.
Fro. 18. — (The v) curves [n = 6)
The E-curve starts at the point m = 3, d = o with a slope — 5/3M
and approaches the singular point {u„ v.) by spiraling around it.
POLYTROPIC AND ISOTHERMAL GAS SPHERES 155
The Z)-curve, as may be verified from our relation of the be-
havior of these solutions as comes from infinity = <» ,
T/ = o) and joins the point {v — 1, u = 0), The 0 -curves all lie in-
side the region bounded by the D-curve, the w-axis, and the part
of the z>-axis o ^ zj ^ i. All these curves approach the point Vs)
by spiraling around it. The ^-curves, which lie outside the D-curve,
arc of the same general nature as in the previous cases. The gen-
eral nature of the (u, z') curves are as shown in Figure 18. This
concludes our discussion of the Lanc-Kmden equation.
22. T/ic isoUicnnal gas spliercr We shall now consider an iso-
thermal gas sphere in gravitational equilibrium. We then have
(368)
where T is assumed to be constant. We can write the foregoing in
the standard form
P = A'p -}- /) ,
where, for the case in hand,
K = T; l) =
till 3
'I'hf f([uati()n <»f crjiiilihrium,
I d fr^ dP\
can be written as
Make the substitutions
p = Xc-'f' ; f =
K_
47 r(/X
t/2
S = a?,
(369)
(370)
(371)
(372)
(373)
STUDY OF STELLAR STRUCTURE
where X is, for the present, an arbitrary constant. Equation (372)
now reduces to
Equation (374), which is our present analogue of the Lane-Emdcn
equation, must govern the density distribution in any region in
which a relation of the kind (369) is valid. If, however, we consider
a complete isothermal gas sphere (or a configuration in which the
central regions are isothermal), we can choose X to be the central
density, in which case ^ = o at f = o. Further, it is clear that
64 / must vanish at the origin. Thus, with the normalization
X = Pc we must find a solution of (374) which satisfies the boundary
conditions
= o, ^ = 0 at { = 0. (37s)
The structure of the complete isothermal gas sphere can be de-
termined when a solution of (374) satisfying the boundary condi-
tions (37s) can be obtained. It does not appear that the equation
(374) can be explicitly integrated, and recourse must be made to
numerical methods. We start the integration by computing the
values of i/' near J = o by means of a power series.
Assuming an expansion of the form
^ + . , . . , (370)
we substitute it in the isothermal equation and determine the co-
efficients a, b, c, . . , , , successively, by equating the coefficients
of the like powers of The first three terms of the series are found
to be
4 '= le - + (377)
A few terms of the foregoing series will enable us to compute 4 / for
^ < I. For ^ > I the solution so obtained must be continued by
standard methods. We shall denote this function by ^(?).
We shall show in § 26 that the complete isothermal gas sphere
extends to infinity. Here we may notice the following formulae for
POLYTROPIC AND ISOTHERMAL GAS SPHERES iS 7
the mass, Af(S), interior to ? and the mean density, p(|), of the mat-
ter interior to which are derived in the same way as the corre-
sponding formulae (Eqs. [67], [77]) for the polytropes
Af(?) = 4 ira 3 X^^ ^ , (378)
P( 5 )=X|^. (379)
Before wc proceed to discuss the nature of the function as
CD and the general solution we shall consider some con-
venient transformations of the isothermal equation.
23. Transfornuilions of the isothermal equation . —
a) Put
1 X .
(380)
Ecpuition (374) takes the form
d^ ^ *
(381)
h) Vui
T
(382)
I'’,(|ualion (374) now lakes the form
•V = 1 '^* ■
(383)
r) We can verify that eciualion (383) is satisfied by Ihe follow-
ing singular solution:
= 2.V'' ;
_ _2
dx A'
(384)
d) Emdens Iramformalion. Because of the existence of the sin-
gular solution (384) we introduce the new varialde 5 delined by
We find
— t// = 2 log .r + s .
“ — ^^5 • 1 --^ ^ 2 d‘'z
dx X (ix ’ (ix^ dx^
(385)
(386)
iS8 STUDY OF STJILLAR STRUCTURE
Equation (383) now takes the form
,d^z . ,
x“ -3— + — 2 = 0.
dx^
(387)
We can eliminate x from the foregoing equation by the transfor-
mation
We have
= I == e* .
dx dt’ dx^
fd^z
dt )
and (387) now reduces to
d^z dz , ,
-2 = 0.
(388)
(389)
(390)
24. The homology theorem for the isothermal equation . — The iso-
thermal equation admits of a constant of homology quite similar
to the homology properties of the Lane-Emdcn equation. In our
present case we have: If is a solution of the isothermal equation ^
then ^(A^) — 2 log A is also a solution of the equation^ where A is
an arbitrary constant.
To show this, write
Tj = y ^ - 2 log .1 . (391)
These transformations lead to an equation in the (i/'*, ry) variables
which is identical in form with the equation in the (^, ?) variables.
Hence, if /(^) is a solution of the original equation, we can choose
as a solution for t/'* the function /(ry). Returning to our original
variables, we now have
= ^*{v) - 2 log ^ = f{A^) - 2 log A , (392)
while /(^) has already been assumed to be a solution. This proves
the theorem.
From (392) it follows that, if we choose for /(f) the function
^(f), then we can derive a whole continuous family of solutions
POLYTROPIC AND ISOTHERMAL GAS SPHERES 159
which arc finite at the origin and which have, further, = o
at ^ = o. We shall denote this family of solutions by and
refer to them as the ‘‘ii-solutions.”
As in the case of the Lane-Kraden equation, we should be able to
reduce the isothermal equation to one of the first order. The vari-
ables introduced in § 23 (case d) enable us to clTcct this reduction.
Introduce the variable y defined by
y =
dz
(it
'J'hen
(/‘Z __ dy __ dy dz dy
dl- ill dz lit ^ dz *
(393)
(394)
Ef[ualion (3()o) can now be written as
(395)
an I'quation analogous to the (y, z) dilTerenlial equation we had be-
fon*. 1'his reduction to a first -order e(|uation is due to the fact that
I lie fuiu'lions y and z arc homology invariant. According to e([ua-
lion (3cS5),
I k‘nce,
- = — ^ -[- 2 !()g $ .
(3</>)
(3<)7)
To show that y and z are homology invariant, we notice that, if
and i;/ .I an* the corresjxinding |)oints along two solutions yf/ and
(which can l)(‘ transformed one into the other by means of a homolo-
gous transformation), then we have*
I) == \[/{0 - log d , (.39^5)
(399)
i6o
STUDY OF STELLAR STRUCTURE
Using (398) and (399), we can easily show that and y*(,^/A)
(defined with respect to the function if'*) are identical with z(J)
and y(J) (defined with respect to the function if).
As another example of the reduction of the isothermal equation
to an equation of the first order, let us consider the functions u and v
defined as follows:
_ je- .
’
(400)
where we have used if'' t.o denote dif/d^. These functions are easily
seen to be homology invariant, and the first-order equation between
u and V can be obtained as follows: We have
£ ^ _ I dip ip"
Since, according to the isothermal equation,
we can re-writc (401) as
or
I du I ,
Similarly, we find that
I dv I
li = ?(“-')•
Hence, combining (404) and (405), wc have
« ^ U — 1
V du u V — ’
Wc shall return to this equation in § 27.
(402)
(403)
(404)
(40s)
(406)
POLYTROPIC AND ISOTHERMAL GAS SPHERES i6i
25. The isothermal 'E-solutions , — ^We shall prove that the solutions
of the isothermal equation which are finite at the origin have necessarily
di^/d^ = 0 at $ = o, and that, consequently, the homologous
family {^(J) } includes all solutions which are finite at the origin.
Consider the isothermal equation in the form (Eq. [381])
g = (x = n) ■ (407)
Such solutions which arc finite at ^ = o correspond to solu-
tions passing through the origin in the (x, plane; as in § 9, we
now have
(f) =ilim[Se-x/f] =0, (408)
\"S/f=o t=o
since x/i = ^ is Unite at the origin, ^ = o; this proves the theorem.
26. The discussion of the isothermal equation in the (y, z) plane . —
I'he functions y and z as we have defined them are homology-in-
variant functions, and consequently each solution curve in the (y, z)
plane corresponds to a complete homologous family of solutions in
the (^, 6 ) plane. In particular, there is just one curve in the (y, z)
plane which corres]>onds to the E-solutions which are included in
the homologous family We shall call the curve which corre-
si)onds to the family 1 '!'($) } the “E-curve” and denote it by yniz).
'Fo rejieal, our eciualions are
Eiirlher,
<lv , .
y - 3/ + -2 = 0,
(409)
+
1
11
(410)
II
1
to
(41 1)
11
■»JO>
II
(412)
, 2 + y - r
(413)
I'Vom (409) we have
i 62
STUDY OF STELLAR STRUCTURE
where we have denoted iyjdz by y'. Substituting for z and y ac-
cording to (410) and (41 1) in (413), we have
, ^
^ w -2 •
We see that the point (o, 2^), where
(414)
= 2 ; 2s = log 2 ,
(41S)
is a singular point of the differential equation (409). There are no
other singular points in the finite part of the (y, z) plane. The exist-
ence of the singular point corresponds to the existence of the singular
solution (384), for by (410) and (415) we have
-h = Zs - 2 log f = log I . (416)
Now the £-curve is characterized by
ypii finite ; 1//' = o as { — > o . (4^7)
From (410), (41 1), and (417) we have
z — CO ; y— >■— 2, y'->o, /-^oo, (418)
Hence, the £-curve touches the line y = — 2 asymptotically as
2^—00. On the other hand, we can show that there cannot be
two solution-curves which are both asymptotic to the line y = — 2
as 2 — CO. For, if there were, let y and y*’' be two dilTerent solu-
tions such that
yr^_-2, y*^— 2, s->— c». (4ig)
We may suppose that y < y* as s — > — 00 . Then we should have
A = y* — 3; > o ; lim A = o .
(420)
From the dilTercntial equation (409) we derive
(lA 2 — £?*
dz
yy^
A .
(421)
POLYTROPIC AND ISOTHERMAL GAS SPHERES 163
Hence,
or by (419)
d log A
= - lim [-
S — > — CO L
<Z log A 1
hm — ^ = -i ,
—CO
A ^ constant (s ~> — ®) , (424)
which contradicts our assumption that A— >0 as s— >— oo. We
have thus proved the uniqueness of the £-curvc.
We shall next examine the behavior of the solution-curves near
the singular ])oint (o, s«). Write
S == Si “f” == Si “h log 2 , (425)
where we now regard s, as small. Kquation (409) now reduces to
y y —2 = 0. (426)
(iZ\
The foregoing equation is exact. Since s, is now considered small,
we can expand the exponential in a power series and retain only
the first two terms. We have, ai)proximately,
y \f: ~ y + = o ; (427)
or, since y dz/dl = dz,/dl, we have, instead of (427),
d’^Zi dz\ . / o\
ddie general solution of (428) can be written as
c, = . , (429)
where A and B are integration constants and r/, and q, an* the roots
of the equation
- q + 2 = o , U^o)
or
(I„(h = l±iWl- («0
STUDY OF STELLAR STRUCTURE
The roots are imaginary, and the solution (429) can therefore be
written in the form
Zi = cos ^ ’ (43^)
where S is a constant. We see that (432) is exactly the limiting form
of our earlier equation (264) as w ^ o, m — > » . From (432) we
have
y = ^ (^ ^ + ^) ~ ^7 sin ^ + ^)] • (433)
We see that the singular point (o, z.) is approached spirally as
< — >■ — ® , ?—>«>. The general run of the solution-curves is illus-
trated in Figure 19.
From (410) we have
— = z — 2 log f = z -f- 2i . (434)
From (425), (432), and (434), we have
—Ip = 2i -H log 2 -h Ae'/* cos < 4 - 5 j ; (435)
or, since { = e~*, we can also write
-'P = log I -I- cos log I - 5
Finally, since
P = ,
we have for the Law of Density Distribution :
A ['^7
TT. cos
p = X-exp
logg-g ($->co). (437)
Since the exponent tends to zero as f , we can further expand
the exponential and retain only the first two terms. We find in this
way that
P = 7^ I { I + |T7» cos log f - sj I ^ 00) . (438)
STUDY OF STELLAR STRUCTURE
1 66
From (438) it follows that as $ 00 the density distribution ap-
proaches that corresponding to the singular solution, namely,
P* = X “ , (439)
asymptotically. The solution (438) intersects the singular solution
(439) at points which asymptotically increase geometrically in the
ratio
I : gar/V? = I : 10.749 (440)
Equation (432) describes the general behavior of the solutions
near the singular point (o, ; hence the law of density distribution
(438) as $ — > 00 is valid quite generally and for the £-solution in
particular.
We have already proved the uniqueness of the /i-curve, which, as
we have seen, becomes asymptotic to the line y = — 2ass— ► — 00,
As / decreases from +00, the E-curve monotonically
rises, intersecting the y-axis at a dciinite point; at this point we
have, according to (410), i/' = 2 log $ or p = X/£“, which is exactly
one-half of the value of p on the singular solution (439^ • Fig- 1 q)-
For further decrease in t the solution approaches the singular point
by spiraling around it; after each revolution s,( = s — z„) and y both
decrease asymptotically in the ratio
I : = 0.09303 (441)
The density distribution as $ — > «»is given by a law of the kind (438).
All solutions other than the E-solution come from y = — 00 , and,
as / 00, they again spiral round the singular point. The behavior
of these solutions as s can also be specified by an apjx^al
to Hardy’s theorem. For, by an application of Hardy’s theorem to
(409), it follows that we should ultimately have one of the following
three possibilities :
dz
-> o ;
dy
dz
dy
dz
I (s _ CO ) . (442)
POLYTROPIC AND ISOTHERMAL GAS SPHERES 167
The first possibility yields the £-curve; the second is dearly im-
possible; hence the only remaining possibility is the third, which
^ y Z Cl (z ( 443 )
where C, is a constant.
From (41 2) and (442) we also have
(y— 00,/— »-t-oo), (444)
y (it
or, integrating, ’ s / \
y = — Ce^ (^-^00,3'— (44s)
where C’ > o is a positive constant. Hence, since y = dz/ dt, we
have
= -cv (446)
at
s = -CV -h C, , (447)
where is a constant. Remembering that ^ = c"', we now have
(?->o). (448)
From (410) and (448), we have, accordingly,
^ = 2 log £ + I - C. (« o) • (449)
'I'he corresponding law for the density distribution is given by
p = \c-* = X ^ (« o) . (450)
which can also l)e written as
p = ^ ‘ (C > o) (« -> o) , (451)
r
where A and (’ are constants. From (451) it follows that along all
solutions of the isothermal equation e.\;cept the /i-solutions and the
singular solution, p— >0 as >0. However, all the solutions have
the same behavior at infinity ; they asymptotically approach the
i68
STUDY OF STELLAR STRUCTURE
singular solution, oscillating with respect to it and intersecting it
at points which asymptotically increase geometrically in the
ratio
27. Discussion of the isothermal equation in the (u, v) plane , — ^Wc
shall conclude our discussion of the isothermal equation by a brief
description of the solution-curves in the (w, v) plane.
Our variables are
2; = ,
(452)
where u and d satisfy the first-order equation
u dv u — 1
V du w + 2) — 3 ‘
(453)
a) The locus of points at which the curves have horizontal tan-
gents is given by
u = 1 ,
(454)
which is a line parallel to the »-axis.
b) The locus of points at which the curves have vertical tangents
is given by
u + v = 3. (455)
c) The two loci (454) and (455) intersect at the point
u, = I ] !), = 2 . (456)
This point of intersection corresponds to the existence of the singu-
lar solution
p
h = log - ;
£
(457)
or
ie-*
'I'
I ;
(45^1)
d) Consider the ^-solutions at f ~ o. It is sufficient to consider
since any other member of the family {^(S) } will lead to the
same (w, v) curve. As J » o, we have, according to equation (,^77),
’^(0 — 6$“ ~ • (I o) . (459)
POLYTROPIC AND ISOTHERMAL GAS SPHERES
169
From the foregoing, we find
g-MO ^ I _ + ^15^4 _ . . .
■ (1 0) I
(460)
(« - 0) .
(461)
Hence, as $ — o.
m = ^ ~ 3(1 - A?')
0) >
(462)
(1 -> 0) .
(463)
Therefore, as ? o, w -» 3 and -4^ o; in other words, the E-curve
passes through the point
UE = 3, m = O (f = o) .
At this point the £-curvc has a definite slope determined by
dVE
(hill. ^
(? - o) ,
or
(464)
(465)
(466)
It is clear, therefore, that the /i-curve starts at the point {u = 3,
V = 6 ) with a negative slope of 5/3 and approaches the point
(u ^ I, V = 2) by spiraling
around it (cf. Fig. 20).
c) All the other solu-
tions also spiral around
this point, and it is clear
that along these curves
V o as u — > . 'rhis
arises because, as we have
already seen, these solu-
tions correspond to a p
which vanishes at ^ = o
and at J = 00 , and hence
must vanish for some finite for this value of z) = o and u — .
/) Finally, we may consider the behavior of the (w, v) curves
20. 'I'lic (/f, v) curves (;/ = “)
170
STUDY OF STELLAR STRUCTURE
near the point (u = = o). Write u - ^ + Ui,
can be written as
3 “t" dv 2 + Wi
V dui + z) *
Equation (453)
(467)
If Ux and > o, we can write approximately
^ dv 2
V dux Ui + v^
(468)
which is of the same form as (348), an equation which arose in a
similar connection when discussing the Lane-Emden equation. We
therefore have (cf. Eq. [354] and replace n by unity)
2;3/a[s(|^ — 3) 4 * 3^] = constant . (469)
Hence, near (u = 3, v = o) the {u, v) curves resemble portions of
generalized hyperbolas asymptotic to the w-axis and the line
S(w - 3) + 3 ^^ = o- (470)
When we put the constant in (469) equal to zero, we get the /i-ciirve
represented near {u = 3, == o) by its tangent (470), thus arriving
at the earlier result (466). This completes our discussion of the poly-
tropic and the isothermal distributions of matter.
28. Composite configurations . — So far we have considered only
complete polytropes. We shall now proceed to a consideration of
composite polytropes, i.e., configurations which consist of dilTerent
zones each characterized by a different value of the index w. Thus,
we can consider configurations consisting of a core of a given index
Wa surrounded by an envelope of another index Wi; in such cases it
is clear that the core will be described by the Lane-Emden function
of index Wa, while the envelope will be described by a solution (in
general not an £-solution) of the Lane-Emden equation of index n..
In this section we shall consider such configurations.
We suppose that we have the equations of state
P = ; P = , (471)
for the envelope and the core, respectively. Further, A', and are
assumed to be constants. It can happen that one or both of them
POLYTROPIC AND ISOTHERMAL GAS SPHERES 171
are “universal” constants. We shall return to this question later,
but for the present we shall formulate the problem in the following
manner:
To construct an equilibrium configuration of a prescribed mass
M and radius R, such that it consists of a polytropic core of index
Hi surrounded by a polytropic envelope of index Wi, it being further
specified that the envelope is to extend inward to a fraction (i — q)
of the radius, R.
Let us first consider the envelope of index w,. The reduction to
the Lane-Eindcn equation of index fh is made by the substitutions
(Kqs. [8] and [10])
, T
p = X.e"' ; P = A'.xr “■ ,
(472)
1 ‘
(473)
Further, we have (Kq. [68])
M (r) == — 47r
(;?, + I)A^
47 r(/
(474)
6 need no longer be the Lane-Eniden function, since a solution
which does not extend to the center can have a singularity at the
origin. X, is, for the present, an arbitrary constant and can be ad-
justed to select any particular solution out of a homologous family
i.e., we can regard X, as the constant of homology. Let d have its
zero at f = £,. Then
M == — 47 r
(Mi + t )/Vi
4 t(/
R =
+ t)A,
u
(475)
I —Hi
(476)
Islimilialiiig X, belwt-i-n (475) and (476), we obtain (cf. Kq. [74])
I
». +• I
4 T
CjO
Ml- 1
Ml
— W|-J_ .<->l |
'“(I'A/” "’ R ’ ,
=
(477)
172
STUDY OF STELLAR STRUCTURE
where, as in equation (337),
As we have already pointed out (§ 21, g), co„, can be used to label
the different solution-curves in any plane in which the Lane-Emdcn
equation reduces to one of the first order.
The problem now is to determine in such a way that the con-
figuration consists of an »j-core occupying a fraction q of the radius.
To do this it is necessary to write down the equations governing
the structure of the core. To avoid confusion we shall use the vari-
ables <f> and 7 ] to describe the core. With the substitutions
I+-
p = ; P = KaK ,
(479)
(480)
we reduce the equation of equilibrium (7) to the Lane-Iimden equa-
tion (in (f> and t]) of index w^. Further, we have
M(r) = — 4ir
(Wa -|- l)jiL3
4x6’
(481)
In the foregoing equations is the constant of homology. Let the
values of the variables at the interface between the core and the
envelope be 6 , 4 >, t). At the interface the values of P, p, r, and
Mir), given by the two sets of formulae ([472], [473], [474] and l47q),
[480], [481]) should be identical. These “equations of fit” are
(482)
A'.X, ' ,
(48,0
(wi -|- i)A i K 2 «, j. _
4x6' ‘ 4ir(/ “ ^ ’
(484)
(w, -I- i)A|
47rG
('fh I y/v j
47rU
a~H 2
(Icj)
dr] *
(485)
POLYTROPIC AND ISOTHERMAL GAS SPHERES 173
We will now show that we can eliminate the constants of homology
Xi and Xa and reduce the system to one involving only the homology-
invariant functions u and v. Raise (484) to the third power, multiply
by (482), and divide by (485). Wc are left with
which can be written as
6 ' ~ ’
m(mi; I) = «(»j: v) ■
(486)
(487)
From (482) and (483) we have
KX/"'B = (488)
Divide (485) by the product of (484) and (488). Wc then have
(«. -t- i) y = {ih + i) ^ , (489)
which can I)e written as
(«. -t- 0 = v) = i>h + i)v{n,-, ij) . (490)
Our equations of lit then are
iiOh \ $) = nOh\ 7 ]) ; K(/7,; rj) , (491)
$ and rj still referrinjj; to the interfaec'.
We will now show how the eciuations (491) enable us to solve the
j)robleni. In § 21 we have already described the nature of the so-
lution-curves in the (//, v) ])lane, The solution-curves in our j)res-
ent (//, r) plane can be obtained very sim])ly by nuiltijdying the
ordinates in the in, v) idane by (w +1). It is therefore clear that all
the characteristic features of the (//, f) ])lane are retained in the
(u, V) plane; in ])articular, the /^-curve separates the region of the
/^'-curves from the region of the A/-curve.
Sinc(‘ the //j-polytroj^ic eejuilibrium extends to the center we can
choose for </> the Lane-Kmden function On, and let X., denote the
central density. Thus, the [//(///, tj), V(n,, rj)] curve to be considered
is the A’-curve. Now the \ K(w,; ^)J= IV curves form a one-
174
STUDY OF STELLAR STRUCTURE
parametric family and some (or all) of these will intersect the E{n-^
curve. A point of intersection between the £(«a) curve and any of
the r„, curves corresponds to a particular solution of the equations
of fit. On the other hand, each point of a («, F) curve corresponds
to a definite value of ^/|i, where defines the corresponding zero
of the solution in the (0, |) plane; the value of f/l, on a solution-
curve in the {u, V) plane is the same for all members of the homolo-
gous family in the { 6 , f) plane which it represents. Hence, the point
of intersection between the E{n^ curve and a r„j curve defines the
value which defines the fraction q. Since this ratio is pre-
scribed, it is clear that only certain of the curves will intersect
the Eiyt^ curve in such a way that the point of intersection will
define for the », envelope an extent equal to that specified. We
select, then, each such solution (there can be more than one solu-
tion), and the value of which labels it is the one appropriate for
use in the mass-radius relation (477). This is the procedure to solve
the equations of fit.
After solving the equations of fit in the manner described, it is
readily seen that the configuration becomes determinate.
Let us assume that we have integrations for all solutions of the
Lane-Emden equation of index n, which passes through some fixed
point ?. = I (say) on the ^axis. A solution of the equations of fit
selects one (or more) of the solutions with certain definite value (or
values) for w„,. This value, substituted in (477), gives /T,, and (476)
now determines the homology constant. A knowledge of w„„
ir„ and will determine the structure of the envelope completely,
and in particular the interfacial density pj.
Also, as we have already pointed out, we can choose p„ = Xj.
Then <l> = d„^ is the Lane-Emden function of index The solu-
tion of the equations of fit provides the value of 17 = at the
interface corresponding to the point on the E(w,) curve through
which the appropriate r„, curve passes. Hence, we have
Pi = Prei‘M '‘^) . (492)
Since pi is already known from the structure of the envelope, p, can
now be determined, thus making the structure determinate.
POLYTROPIC AND ISOTHERMAL GAS SPHERES 175
The procedure of solving the equations of fit becomes slightly
altered if, instead of the extent (i — q), the constant K, is assigned
some definite value. In this case, equation (477) determines
and therefore the particular T,,. curve. We must find whether this
curve intersects the £(mj) curve. If it does not, then a composite
configuration of the character contemi>lated is impossible. If, how-
ever, the r„, curve intersects the curve, then the equations
of fit have a solution and the point of intersection will determine the
fraction of the radius occupied by the core. The following theorem
is of interest in this connection.
If [u(n,; ■r}),V(n.-,ri)] corresponds to the E(nj) curve, then the equa-
tions of Jit have a solution if, and only if, the n, -envelope is described
by an F- or an M.-solulion accordinj^ as iij is less than or greater than
n,. Further, it is assumed that n, < 5, n^ < 5, and n, n..
From our discussion in § 2 1, it is clear that to prove the foregoing
theorem we have only to show that the A(«j) curve lies entirely be-
low or above the K(n,) curve according as n, is greater or less than w,.
Since the ordinates in the (u, v) ])lane are increased by the factor
(n + 1) to transform to the in, F) ])lane, it is clear that the start-
ing-slope of the Ji(n) curve is given by d'h]. [329]),
= _S
diiE 3
( 49 .^)
C’onse(iuently, the FUh) curve lies initially below the F{n,) curve if
Similarly, the Fin.) curve lies initially above the Fin,) curve if
n. < It,. I'Vom our discussion in § 21, it follows that, if n, < 5,
then an Fin.) curve lies entirely below (or above) an Fin,) curve if
it lies initially below (or above) it.
Mnally, since (he A/-curves lie entiri-ly behiw the /'J-curve while
the F-curves lie entirely above it, it follows that the iCin.) cuive
intersects all the A'/-curves belonging to n, if Mj > n„ while the
176
STUDY OF STELLAR STRUCTURE
E(n2) curve intersects all the J^-curves belonging to fit if Wa < Wi.
This proves the theorem.
For > 5 the arrangement and the character of the solutions
become different and the enumeration of the different possibilities
becomes more complicated. Nothing new in principle, however,
arises.
BIBLIOGRAPHICAL NOTES
As has been pointed out, the notion of convective equilibrium is due to Lord
Kelvin {Mathematical and Physical Papers, 3, 255-260, first published in 1862),
whose investigation may properly be described as the real forerunner of the sub-
sequent studies by Lane, Ritter, and Emden. In view of the fundamental char-
acter of the work of these authors we shall describe in some detail the actual
contributions of each of them.
I. J. Homer Lane, Amer. J. Sci., 2d ser., 50, 57-74, 1869. Lane’s paper,
“On the Theoretical Temperature of the Sun under the Hypothesis of a Gaseous
Mass Maintaining Its Volume by Its Internal Heat and Depending on the Laws
of Gases Known to Terrestrial Experiment,” considers for the first time the
equilibrium of a steUar configuration. It should, however, be pointed out that
the problem of the gravitational equilibrium of a gas sphere is considered only
incidentally and that the “theoretical temperature” refers to the surface tem-
perature of the sun. Lane’s principal object in his investigation was to determine
the temperature and the density at the surface of the sun. In order to deter-
mine these quantities, he adopts the following procedure. From the value of
the solar constant as known at that time (Herschel and Pouillet’s determina-
tion), he attempts to derive the surface temperature. Stefan’s law was still un-
known (Stefan published his law in 1870), and Lane therefore uses certain ex-
perimental results of Uulong and Petit and of Hopkins on the rate of emission
of radiant energy by heated surfaces. Using the empirical law derived by Hop-
kins, Lane estimates the surface temperature of the sun to be 54,000° 1^' or
30,000° Kelvin. Lane realizes that this estimate depends on a gross extrapola-
tion of the experimental results, but it is interesting to note that, in principle,
his method is not different from modern methods of determining the elTcctive
temperatures from Stefan’s law.
Lane’s next problem is to determine the density corresponding to his derived
surface temperature. To this end he solves the equilibrium of the sun as a whole.
Assuming that the sun is in “convective equilibrium,” according to the then
recent ideas of Lord Kelvin and referring to the work of Clausius, Lane argues
that, because of the “fierce collisions of compound molecules with each other at
the temperatures supposed to exist in the sun’s bexiy, their component atoms
might be torn assunder” it would be safe to assume for the ratio of the specific
heats the value for a monatomic gas namely 7 = 5/3. (Lane also considers,
formally, the case 7 = i-4*) The mathematical problem of the equilibrium of
POLYTROPIC AND ISOTHERMAL GAS SPHERES 177
a configuration under its own. gravitation with an underlying law P ^
is formally solved and the appropriate “Lane-Emden’’ function numerically
isolated. He gives the graphs of p and T as functions of the radius vector. From
these graphs he finally reads off the value of p corresponding to T = 30,000°
and obtains p = 0.00036. When the crudeness of the then available data is
considered, Lane’s success in estimating p and 7 " at the surface of the sun is a
remarkable achievement. It may thus be said that his paper has made him the
author not only of the first investigation of the physical conditions in the solar
atmosphere but also of the first investigation on stellar interiors though the
latter was not his primary concern.
Lane’s paper contains no explicit reference to what is generally called “Lane’s
law.” Indeed, he does not consider homologous transformations at all — it was
Ritter who first considered such transformations explicitly; the reason the law
;• 2 '(:r) = constant (for a uniform expansion or contraction of a gaseous con-
figuration) is called “Lane’s law” is explained in Simon Np:wcomb’s The Rc 7 m-
nismiccs of an Aslrononicr (1003, Cambridge, Mass., U.S.A.). The following is
an extract from Newcomb’s book (pp. 245 -240):
After the paper in question appeared I called Mr. Lane’s attention to the fact that
I did not find any statement of the theorem which he had mentioned to me to be con-
tained in it. He admitted that it was containe<l in it only impliviily and proceded to
give me a brief and simple demonstration. So the matter stood until the c.entennial
year i(S76 when Sir William 'J'homson paid a visit to Ibis country Among other
matters I mentioned this law originating with Mr. J. Homer I.ane. He did not think
it could be well foun<led and when 1 attempted to reproduce Mr. J.ane’s verbal demon-
stration I found myself unable to do so When 1 again met Mr. Lane I told him
of my dilliculty and asked him to repeat the demonstration. He did so at once and I
sent it olT to Sir William. 'Phe latter immediately accepted the result and published a
paper on the subject in which the theorem was made j)ublic for the first time.
Newcomb concludes:
Altogether I feel it eminently appropriate that his name should be perpetuated by
the theorem of which 1 have spoken.
'Phrcc points of historical interest should be noted: (i) Lane’s interest was
primarily wdth the solar atmosphere; (2) his interest in the gravitational equilib-
rium of a gaseous configuration was incidental to the main object of his pub-
lished investigation; and, finally (3), lane mu.st have derived the law asso-
ciated with his name essentially from an argument involving the homology in-
variance of the e(iuilibrium configurations built on the law P oc in
Kki.vin’s laper, /V// 7 . A/</g., 22, 2cS7, 1SS7, the homolog)- theorem (as we have
proved it in § S) is e.xplicilly i)roved; and since, further, Kelvin’s paper con-
tains a reference to Lane’s i)aperand also to a letter from Newcomb, it is clear
that Newcomb’s reference to Kelvin’s paper as making “public for the first
time” Lane’s law mu.st refer to Kelvin’s proof of the homology theorem.
11 . A. RriTKK. Ritter’s investigations are very remarkable in their range
178
STUDY OF STELLAR STRUCTURE
and depth; through his papers he shows himself to be a pioneer of very great
originality. Unlike Lane, Ritter was primarily interested in the equilibrium of
stellar configurations, and his contribution to the formal mathematical theor\'
is so great that such aspects of the theory of gaseous configurations built on
the law P oc as are commonly known are almost entirely due to Ritter.
It should be noted further that Ritter's work was all done independently and
without knowledge of Lane’s paper. Ritter’s studies extended over a period of
six years, and his eighteen communications on “Untersuchungen liber die
Hdhe der Atmosphiire und die Constitution gasformiger Weltkdrper” presented
to the Wiedemann Annalcn during the years 1878-1889 form a classic the value
of which has never been adequately recognized, though Emden refers very en-
thusiastically and at great length to the wealth of material that is contained in
Ritter’s work. The following is a list of Ritter’s papers; the most important of
them are starred (*), and the essential results contained in them are briell>'
reviewed.
I. Si 40S) 1878 *8. II, 332, 1880 14. 17, 332, 1882
*2. 5, 543, 1878 *9. II, 978, i88o 15. 18, 488, 18S3
*3- 6,135,1878 10. 12,445,1881 *16. 20,137,1883
4 * 7 » . 304 ) II- 13. 3fio, 1881 *17. 20,897,1883
*5, *6. 8, 157, 1880 12. 14, 6io, 18S1 j8. 20, 910, 1883
7. 10, 130, 1880 *13. 16, 166, 1882
In (2) the uniform expansion and contraction of gaseous configurations are
considered, and Lane’s law (independently of Lane) explicitly proved. 1'hc cos-
mogenetic equation of state is here defined, and what we have called “Rillcr’s
theorem” (chap, ii) is also proved in this paper.
In (3) the fundamental differential equation for n = 2.44 is established, and
the appropriate “Lane-Emden” function obtained. This paper also contains the
derivation of the Helmholtz- Kelvin time scale (cr. chap. xii).
In (5) and (6) the equation il = -3/A/F is obtained (his Kq. liS6|) and
what we have ciilled “Ritter’s relation,” namely,
U= ^ r S2 ,
aCT- - 1) ’
is also obtained (his Kq. [igo]). In this paper the adiabatic ])ulsation of a gas
sphere is considered for the first time, and the fundamental result is i)ro\'ed
that (7 = 4/3) separates the configurations which are stable (7 > 4/3) from
those which are unstable (7 < 4/3)- Ritter also proves the important result
that the period of oscillation of a gas sphere is inversely proportional to tlu‘
square root of its mean density. It should be noted that Kilter dcvelojis the
theory of pulsating configurations with a definite view toward a theory for the
variable stars.
In (8) Ritter establishes (explicitly for the first time) the fundamental dilTer-
ential equation governing the structure of gaseous configurations with an under
lying law P a nia equation (295) is what we have called the “Lane-
POLYTROPIC AND ISOTHERMAL GAS SPHERES 179
Emden” equation, though it should have been more appropriate to have called
it the “Lane-Ritter** equation. The “Lane-Emden” functions for w = i, 1.5,
2, 2.44, 3, and 4 are obtained — for 11 — i, he uses = sin This paper also
contains proofs of the important formulae
3 GM^ I GM^
“= "s -« i? : ^ “ (57-6) /? •
Ritter also discusses the importance of the case n = 5.
In (9) Ritter considers composite configurations consisting of incompressible
cores and gaseous envelopes. In this connection Ritter draws attention to the
importance of the solutions of the fundamental differential equation other than
those which are finite at the origin. In particular he uses the general solution
Q = .1 sin — 5)/^ when considering the case n = i. Ritter was thus not
only the first to consider composite configurations but also the first to recognize
the importance of solutions which have a singularity at the origin.
In (13) Ritter considers the isothermal gas sphere and isolates the singular
solution = 2/iK In this paper he also proves the integral theorems which
are referred to in the bibliographical note for chapter iii.
In (t6) and (17) what is generally called the “giant-dwarf” theory of stellar
evolution was originated and considered for the first time.
Krom this very brief and inadequate summary of the important results that
arc contained in Ritter’s papers, it should be clear that almost the entire founda-
tion for the mathematical theory of stellar structure was laid by him. Jlis
papers contain, in addition, discussions of a variety of both stellar and meteoro-
logical phenomena which arc beyond the scope of our present note.
III. Lord Kklvint. It is somewhat surprising that twenty-five years should
have elapsed before Lord Kelvin applied his idea of convective eriuilibrium to
the study of gaseous configurations. Ilis paper in the Philosophical Magazine in
1887, to which we have already referred several times, is still of interest because
of the very short space in which he (independently of his predecessors) derived
many of the essential results. It is interesting to recall that Kelvin’s interest
in the problem “of the equilibrium of a gas under its own gravitation only”
originated in a question set by P. G. 'Fait in an examination paper (Kerguson
Scholarship Examination, Glasgow, October 2, 1885). 'fait’s question reads:
Assuming Hoyle’s Law for all pressures form theeiiuation for the eciuilihrium-density
at any distance from the centre of a spherical attracting mass, placed in an infinite
space filled originally with air. l^'ind the special integral which di'pends on a power of
the distance from the centre of the sphere alone.
In his 1887 paper Lord Kelvin promises a further pa|)cr, but actually he re-
turned to the subject only twenty years later in his posthumous pajicr on “'fhe
Problem of a Spherical Gaseous Nebula” {Collected Papers^ 5, 254 283), which
appeared in ic)o8. 'Phis last [lapcr contains an extremely attractive summary of
the state of the subject prior to the publication of Emden’s book. Finally, it
i 82
STUDY OF STELLAR STRUCTURE
3. E. A. Milne, 91, 4, 1930; 92, 610, 1932. In these papers Milne in-
troduces the variables u and which are particularly suited for the discussion
of composite configurations. His method is largely used in § 28.
4. S. Chandrasekhar. The author has included in this chapter several of
his results on the Lane-Emden equation. The discussion in §§17 and 20,
where Hopf’s methods are generalized to cover the cases « > 3, is mostly new.
The D-solutions with the behavior 6 C/^ as $ — > » are isolated here for the
first time, as are also the grid properties of these D-solutions. The complete dis-
cussion in the (w, ii) plane (§21), the introduction of the (w, v) variables for the
isotliermal gas sphere (§ 24), most of the discussion of §§ 25 and 26 [with the
exception of the part dealing with the derivation of equation (438) describing
the behavior as ^ which is due to Emden], and the results contained
in § 27 are all new.
5. H. N. Russell, M.N., 91, 741, 1931.
6. N. Fairclough, M.N., 91, 62, 1930; 92, 644, 1932; 95, 585, 1935. These
papers contain the tabulation of the general solutions for « = 3 and n = 3/2.
CHAPTER V
THE THEORY OF RADIATION AND THE
EQUATIONS OF EQUILIBRIUM
We have already shown in chapter ii, by an application of the
laws of thermodynamics, that the energy density u of black-body
radiation at temperature T is proportional to the fourth power of the
temperature (Stefan’s law). In this chapter we shall be concerned
with a further discussion of radiation problems and the bearing of
these problems upon an understanding of the physical conditions
that could be encountered in stellar interiors.
I. Fundamental notions and dejinitions . — We shall begin with a
few delinitions:
a) The spccijic mtcnsily of radiation at a f;iven pointy P, and in a
given direction. Let da be an arbitrarily chosen small clement of
surface containing the point P. At a given instant of time there will
be rays' traversing this element in all the dilTerent directions. Let
us consider a speciiic direction say the .v-direction. Through every
point of da construct cones abutting on da having axes parallel to
the .v-direction with solid angles at the apex all equal to a definite
inlinitesimal amount doj. These cones deline a semi-infinite volume
in the form of a truncated cone.^ The energy in the form of radiation
traversing the element of area da and in the semi-infinite volume
defined, during an interval of time dt, can be written as
T cos 0 dadoidt , (i)
where 0 is the angle which the ,v-direction makes with the normal
to da. The quantity thus introduced, depends naturally on the
position of the point /^ the direction .v, and (if the state is nonsta-
tionary) on the time t. I is said to define the specific intensity of ra-
diation at the point P and in the prescri1)ed direction.
A radiation field is said to be isotropic if 1 depends only on the
‘ In llu' of ^geometrical optics.
^'rhe construction used here is said to (lefinc a “pencil of radiation.”
i84
STUDY OF STELLAR STRUCTURE
position of the point P and is independent of direction at P; if,
further, I is independent of the position of the point P as well, then
the radiation field is said to be homogeneous and isotropic.
The ^-direction can be completely specified by the angle d{o ^
6 ^ tt), which we have already defined, and the ''azimuth” <t>
(o ^ 0 ^ 27r). The element of solid angle Jco, defined by the ranges
(0, 6 + dB) and (0, 0 + d<t>)^ is
doa = sin B dBd4> ; (2)
and the expression for the energy traversing the area dc in the di-
rections confined by element of solid angle doj (0, 0 + 0, 0 + d<i>)
during a time dt is, then
/ sin 0 cos 0 d6d(l>dcdt . (3)
b) The flux of radiation , — The total amount of radiant energy
traversing the surface element da from one side to another, expressed
in terms of unit area and unit time, can be written as
n ir/2
I sin 0 cos 0 dBd(t > . (4)
In the same way, the amount of radiant energy traversing da in
the opposite direction, expressed also in terms of unit area and unit
time, is given by
F- = — I i I sin 0 cos 0 dddij ) . (5)
Jir/a
The net flux of radiation, F, across da per unit area and unit time is,
therefore,
F = F+-F., (6)
or by (4) and (s)
or, again, by (2)
F =
I sin 0 cos 6 d 6 d(t > ,
F =fl cos 8 dw ,
(7)
( 8 )
where the integral is extended over the complete sphere.
RADIATION AND EQUILIBRIUM
185
If we consider a Cartesian system of co-ordinates, (X, F, Z), and
denote by Fy, and Fs the net fluxes at a point across elements of
surfaces normal to the directions X, F, and Z, respectively, then we
should have
Fx = J/Wo) ; Fy — jlmdo) ; Fs = jlndta , (9)
where I is the specific intensity at the point under consideration
and in the direction specified by the direction cosines Z, w, and n.
If we consider the flux across any surface da, the normal to which has
the direction cosines /i, Wi, and then we should have
Fi^, y, 2 ; Z, m, n) cos ^ Jco , (10)
where ^p is the angle between the direction (Z, m, n) and the direc-
tion (Z,, m,, w,); hence,
cos yp — lli + mmi + ntii . (ii)
By (9)7 (^o)> 'wc now have
Fl^, = hFx + + '«iEa . (12)
In other words, we can regard the flux as a projection of a vector
which has the components Fxy Fy, and Fs in the three principal di-
rections.
c) Distribution in the frequency of radiation. specific inten-
sity which is related to the total energy radiated in a certain direc-
tion can be further divided into the intensities of the radiations in
the different frequencies which travel independently of one another.
If we consider an infinitesimal interval {v, v -f dv), then the specific
intensity h is so defined that the total energy in the frequency in-
terval {v, V + dv) which crosses an element of area da in a direction
making an angle 6 with the normal to da and in an element of solid
angle do), is, during a time dt,
ly cos 6 dad(jodtdv . (13)
Strictly speaking, wc can never consider a rigorously monochromatic
pencil of radiation. It is always necessary to consider a nonzero,
though infinitesimal, frequency interval.
i86
STUDY OF STELLAR STRUCTURE
From our definitions it follows that
Ivdv = I . (14)
We shall refer to I as the “integrated intensity,” in contrast to the
monochromatic intensity, Ip,
In the general theory of radiation we have to distinguish, further,
the different states of polarization of the radiation, but in the appli-
cations that we shall consider it is not necessary to go into these
finer details.
d) The amount of radiant energy flowing from one element of surface
to another element of surface . — ^As the treatment of this problem is
the same for the integrated intensity, /, as for the monochromatic
intensity, we shall explicitly consider only the former case.
Let da and da' be the two elements of surface surrounding the
points P and P', respectively. Let r be the distance between P
and P'. Further, let PP' make angles 6 and 6 ' to the directions of
the normals to da and da' at P and P', respectively. Finally, let
I be the specific intensity at P in the direction PP'.
In free space the energy which traverses the element da in time
dt and which also traverses da' is, according to our definition of
intensity,
dE = / cos B dffdcodt , (15)
where do) is the solid angle which the element da' makes at P. "I'his
is seen to be
, da' cos B' , .
dco = . (16)
From (15) and (16) we have
^ cos 6 cos B' dada' ,, . .
dE = I dt. (17)
An immediate corollary of the foregoing result (17) is that the s/)c-
cific intensity is constant along the path of any ray in free space.
For, if dE' is the energy which traverses the element da' and
which also traverses the element da, then, according to (17),
(, 8 )
RADIATION AND EQUILIBRIUM 187
where /' is now the intensity at P' in the direction PP'. But it is
clear that
dE = dE'. (19)
Comparing (17) and (18), we see that / = /'; we thus have
_ cos 6 cos 6 d(Td(T
dE == dE' = / ^ dt
(20)
We see that equation (20) is symmetrical between the unprimed
and the primed quantities and exhibits in this sense a certain reci-
procity; equation (20) is in fact a special case of a more general
reciprocity theorem.
e) The energy density oj radiation at a given point . — The energy
density, w, of the integrated radiation at a given point is the amount
of radiant energy per unit volume which is in course of transit, per
unit time, in the neighborhood of the point considered.
Consider a point P, and construct around it an infinitesimal ele-
ment of volume the bounding surface of which we shall denote
by (T. We shall further restrict the surface o* to be convex every-
where. To allow for all the radiation traversing v, we surround o*
by another convex surface 2 such that the linear dimensions of S are
large compared with the linear dimensions of c; nevertheless, we can
arrange, at the same time that the volume element inclosed by S is
still so sufficiently small, that we can regard the intensity in a given
direction as the same for all the points inside 2.
Now, all the radiation traversing the element v must have crossed
some element of the surface iS. Let be such an element. The
energy flowing across dS which also flows across an element d<r of a
per unit time is, according to (20),
T cos Q cos 0 dad'^ , .
I , (21)
where 0 and 0 are the angles which the normals to da and dli
make with the radius vector r which connects the two elements. Let I
be the length traversed by the pencil of radiation considered through
the volume element v. The radiation incident on will have trav-
ersed the element in time l/c, where c is the velocity of light.
i88
STUDY OF STELLAR STRUCTURE
Hence, the contribution to the total amount of radiant energy in
course of transit through the volume element v by the pencil of
radiation considered is
cos 6 cos 0 dadh I
But the volume dv, intercepted by the pencil of radiation from the
element v, is
dv = 1 cos B d(T .
(23)
Hence, we can write (22)
as
c '
(24)
where
, cos 0 dS
aoy =
( 2 S)
is the element of solid angle subtended by dS at P. Therefore, the
total energy in course of transit through the volume element v by
radiation from all directions is
(26)
where the integration with respect to co is extended over the whole
sphere. Hence,
(27)
since the energy density is expressed in terms of unit volume.
We can similarly define the energy density ti,dv of the radiation
in a specified frequency interval (v, v + dy). We have, as before.
If the radiation is isotropic.
RADIATidN AND EQUILIBRIUM
189
/) The emission coefficient . — Let us consider a small element of
mass m which is radiating. Let us further consider the radiation
emitted in the directions specified by an element of solid angle dca
and in a definite frequency interval {v, v + dv). The amount of ra-
diant energy emitted in the element of solid angle in time dt and in
the frequency interval {v, v + dv) can be written as
jvmdoydtdv . (30)
The quantity thus introduced, is called the “emission coefficient
for frequency v'" It should be remarked that, even if the element of
mass is isotropic, it does not necessarily follow that the emission of
radiation takes place uniformly in all directions. As we shall see
presently, a further necessary condition for the emission of radiation
to be uniform in all directions is that the element of radiating mass
should itself be in an isotropic field of radiation.
If we consider the emission in a definite frequency I'nm, correspond-
ing to a quantum transition between two definite states, m and n, of,
the atoms forming the medium (the states need not be discrete
states), then, according to the Bohr frequency condition,
hvnm = En — E,„ , (31)
where E,, and E^ are the corresponding energies of the two stationary
slates. Emission in the frequency Vnm lakes place because in a given
instant of lime there will be a certain number of atoms in the excited
state n, and when these atoms jump to the state m, they will emit
quanta of energy ////„„,. Quantitatively, the emission of radiation in
the frequency Vnm is determined by the Einstein coefllcients and
Bnm of si)ontancous and induced emission, respectively. These
coclficients are defined as follows: The probability that in an in-
terval of time dt an atom in the excited state n emits a quantum
of energy hPnw, in the directions confined to an element of solid
angle dca and in the absence of an external field of radiation, is
Anmdcadl. This spontaneous emission takes place uniformly in all
directions. The probability of the emission of a quantum is in-
creased if the atom in the state n is exposed to a field of radiation of
frequency Pnm- We take account of this induced emission by intro-
jgo
STUDY OF STELLAR STRUCTURE
ducing the coefficient Bnm; it is defined in such a way that the prob-
ability that an excited atom in state n is stimulated by an external
field of radiation to emit a quantum hvnm in the directions specified
by an element of solid angle in time dt, is given by
Bnml > ^3 2 )
where is the intensity of the radiation of frequency Vnm at the
point where the atom is located and in the direction defined by doj.
The expression (32) for the probability of induced emission arises
because the emission of radiation induced by a given pencil of radi-
ation takes place in exactly the same direction as the incident pencil.
Hence, the total probability per unit time of induced emission is
. (33)
Thus, the total emission of energy by one single atom in the state
n per unit time is given by
hPnm[. 4 '^-^nm “1“ . (, 34 )
Finally, if there are Nn atoms per unit volume in the state n, wc
have
Nn
+ BnmL^^JhVnmdu) , (35)
where p is the density. From (35) we see that an element of mass
radiates uniformly in all directions only if it is in an isotropic iield of
radiation.
The total emission in all directions per gram of material is given by
• (36)
g) The absorption coefficient , — A pencil of radiation traversing a
medium will be weakened by absorption. If the specific intensity
Ip of radiation at frequency v becomes Ip + dip after it has traversed
a medium of thickness ds, we can write
dip ^^Kpplpds •
(37)
RADIATION AND EQUILIBRIUM
191
It should be remarked that h + dL is the intensity of the emergent
radiation which is in phase with the incident radiation. The quan-
tity fCp so introduced is defined as the ^^mass absorption coefficient’^
for radiation of frequency v.
From (37) we find, on integration, that
7 ,(5) = (38)
where IX^) is the intensity after the radiation has traversed a length
^ of the medium. Equation (38) is generally written in the form
L{s) = , (39)
where
The quantity Tp is called the ^‘optical depth” of the material trav-
ersed to radiation of frequency v.
If we consider the case of absorption between two stationary
stales n and m as in section /above, then the absorption of radiation
of frequency Vnm arises from the excitation of the atoms from the
lower state m to the higher state n. We express this quantitatively
in terms of the Einstein coelficient of absorption, defined in
such a way that the probability of an atom in the state w, ex])osed
to radiation of frecpiency absorbing a quantum //J'wmin timed/,
is given by
(IL , (41)
where the integral is extended over the complete sphere. 'I'he rela-
tion of the coelficient B,nn to the mass absorption coelficient is
easily seen to be (cf., sec. //, below)
BmJlVn
(42)
where N is the number of atoms in unit volume in the state w and
p is the density.
//) Total ahsorplion. Consider a small element of mass m which
is exposed to a field of radiation, d’hen in order to calculate the
192
STUDY OF STELLAR STRUCTURE
total absorption of radiant energy in the frequency v per unit time,
we inclose the element of mass by a larger surface 2 outside the
bounding surface of m which we denote by <r; the linear dimensions
of (T are taken to be much smaller than those of 2 . Then, proceeding
as in the calculation for the energy density, we have for the amount
of energy traversing an element of surface di: of 2 in unit time, and
which is incident on an element da of the bounding surface of w,
, cos 6 cos 0 dffdl^ , , .
;; dv , ( 43 )
where we have used the same notation as in section e above. Of the
amount of energy (43), the amount absorbed by the element of
mass is obtained by multiplying (43) by K,pl, where I is the length
intercepted in m by the pencil of radiation under consideration.
Hence, the amount of energy absorbed per unit time from the pencil
of radiation under consideration is
where
- cos 6 cos 0 dadX , , r 1 i ,
Iv Kppldp = K^LaQ)dmdv ,
dm = pi cos 9 da
cos 0 dX
Hence, the total energy absorbed is obtained by integrating (44)
over m and co. We thus find that
Kpfndvf Ipdcj ( 47 )
specifics the amount of energy absorbed by the element of mass con-
sidered from the radiation field in the frequency interval {p, p + dp).
i) The pressure of radiation . — The existence of light pressure fol-
lows from Maxwell’s electromagnetic theory of light. It also follows
from the quantum theory, according to which a quantum of energy
hv is associated with a momentum hpje (where c is the velocity of
light) in its direction of propagation. From this it follows that ra-
RADIATION AND EQUILIBRIUM
193
diant energy of amount E traversing a medium in a specific direc-
tion carries with it a momentum E/c, the momentum exerted being
in the same direction as the pencil of radiation.
To calculate the pressure of radiation at a given point P, we have
to consider the net rate of transfer of momentum normal to an ar-
bitrarily chosen element of surface da containing P.
If we consider radiation of frequency v as incident on the surface
da and making an angle 6 with the normal to da, the amount of
radiant energy in the frequency interval (v, v + dv), traversing
da in directions specified by an element of solid angle doi in time dt, is
Tv cos B dmlvdadt . (48)
This amount of radiant energy carries with it the momentum
- Jv cos B doidvdadt
c
(49)
in the direction of I,. Hence, the normal component of the momen-
tum transferred across da by the pencil of radiation under consider-
ation is
- dadl If cos“ d dia dv . (50)
c
Therefore, the net transfer of momentum across da by the radiation
in the frequency interval (v, v + dp) is
dadt Iv cos- 0 doj dv , (51)
where the integration is to be carried over the complete sphere.
Since the pressure at a point P is defined as the net rate of transfer
of momentum normal to an arbitrarily chosen infinitesimal element
of surface containing and expressed in terms of unit area, we can
write for the pressure {pr{y)dv\ due to radiation in the frequency
interval {v, v dv)\
priy) = ~ ^ ^ Iv COS'* B sin B dBd<t) . (52)
194
STUDY OF STELLAR STRUCTURE
If the radiation is isotropic, we have
pr{v) = 2ir - I cos* 6 sin 6 dQ = — rly . (53)
cjo 3c
Comparing this with (29), w'e have
Pr(v) = .
(54)
We can define the integrated radiation pressure, pr, due to radiation
in all the frequencies, by
or by (52)
pr = J pr{v)dv ,
I cos^ d do 3 ,
(5S)
(56)
where I now defines the integrated intensity. For isotropic radia-
tion we have
pr = iu, (57)
a result we have already used in chapter ii, § ii, to derive Stefan’s
law.
j) The pressure tensor , — Let us consider an element of surface
normal to the -direction. The rate of transfer of the :x:-component
of the momentum across the element of surface (per unit area) by
the radiation confined to an element of solid angle dco, about a di-
rection whose direction cosines are /, m, and w, is
j Ildu I . (58)
(We are considering integrated radiation but the treatment is
equally valid for monochromatic radiation; we need only to replace
I by lidv.) The total rate of transfer of the :»c-momentum across the
element per unit area is, then,
iJ/Wa,. (S9)
RADIATION AND EQUILIBRIUM
IQS
The foregoing quantity defines the o^-component of the pressure ex-
erted across the element under consideration. We Write it as pxx
(strictly speaking, we should have a suffix r for radiation, but this
would unnecessarily burden the notation).
In the same way, the y- and the s-components of the pressure
across the element of surface considered are
= Ilmdo) ; px2 =
Ilndco .
Similarly, by considering elements of surface normal to the Y- and
the Z-directions, we can define the further sets of quantities (pyx,
pyy, pyz) aud (p,x, pzy, • Thc nine quantities we have thus de-
fined are said to form the “stress tensor”:
Pxx
pyx
Pxv
^ Ilmdo) ;
^ . T
Tlndu ,
; Imldo) ;
pyy
P Im^dw ;
I 1
f- - ;J
^ Imnd(ji , •
Inldo} ;
P-y
^ I nnido) ;
J J
! that
P^v =
Pyj^ i
P
xz = pzx ’)
II
in other words, thc tensor is symmetrical. The mean i)ressure p is
defined l)y
p = Kpxx + pyy + A.c) . (^>3)
rVom (6i) it follows (since + nf + = 0 that
/(/w = In j
a relation which is generally true.
If the radiation is isotropic, we have
p — pxx — pyy — psz “ 3 ^-^
196
and
STUDY OF STELLAR STRUCTURE
pxy Pvx O ) pxx — Pzx ~~ O j pyx — psy — O . (66)
Whenever relations (65) and (66) are true, we say that the stress
tensor reduces to a simple hydrostatic pressure.
There is another simple case in W;hich the system of stresses (61 )
reduces to a simple hydrostatic pressure, namely when
W=0O
/ = /o + , (67 )
(a+/3+7=2«+i)
» = I
where /» and the “coefficients” hay are all arbitrary functions of
position only. The triple suimnation in (67) is extended over all
possible sets (a, ;S, y ^ o) such that a + j8 + 7 is odd. From (67 )
and (61) it follows that
47r
Pxx “ pyy “ pzz ” ioj pxy ~ Pyx “ O j CtC.
k) The mechanical force exerted by radiation, — To determine the
mechanical force exerted by radiation, consider a thin cylinder of
cross-section da and length ds in the direction normal to da.
The amount of radiant energy in the frequency interval {y,,
V + dv) incident on da, in the directions specified by an element of
solid angle dco, about a direction making an angle 6 with the ^-di-
rection, and in unit time, is
L cos d d(xd(jodv , (69)
The amount of this energy which is absorbed in the cylinder is
Iv cos 6 dadtadv Kvp sec 6 ds , (70)
since sec 6 ds is the length of the path intercepted in the cylinder by^
the pencil of radiation under consideration. ^ The amount of momen-
tum thus communicated in the direction of 1 ^ is obtained by divid-
ing (71) by the velocity of light, c. The normal component of the
3 For the validity of (71) (to the first order) it is necessary to assume that ds is of
a higher order of smallness than da. This assumption is also made in § i (/e) and § 2.
RADIATION AND EQUILIBRIUM
197
momentum thus communicated to the cylinder by the pencil of
radiation under consideration is given by
Iv cos 6 dadoidp kpP sec B ds - cos 6 .
(71)
To obtain the normal force per unit area, we have to divide the fore-
going by da. Finally, integrating over all the directions of the in-
cident radiation, we obtain for the mechanical force per unit area of
a cylindrical slab of thickness ds:
Kvpds
c
'f
Ip cos 6 do3 dv ,
(72)
The foregoing force per unit area in the ^-direction, on the cylindri-
cal slab considered, arises from absorption.
We shall now examine the possibility of there being some additional
mechanical force arising from emission. The spontaneous emission
which takes place uniformly in all directions will not give any net
resultant force. On the other hand , the induced emission which takes
place in exactly the same direction as the incident stimulating radia-
tion will give a net resultant if the incident field of radiation is not
isotropic. From equation which gives the atomic probability
of induced emission, we obtain for the normal component of mo-
mentum communicated to the cylinder by the emission induced by
the pencil of radiation defined in (70) the exiiression
BnmhPnnJr^^jivdoJ p(l(T(ls - COS 0 {v„
= p).
(7,0
We have the negative sign in (7^0 because the emission takes place
in the forward direction and corresponds to a loss of momentum by
the infinitesimal cylindrical slab considered.
Using (42) for our definition of the absorption coefllcient and in-
tegrating (7,0 over all the directions, we obtain for the normal force
per unit area on the slab considered and in the ^-direction:
^ nBnni
N B
COS 6 (icj dp
{Pnm
= 0 . (74)
STUDY OF STELLAR STRUCTURE ■
198
Combining (72) and (74), we have for the net normal force per unit
area acting in the ^-direction on a cylindrical slab of thickness ds:
- Kf ( I — pds F,{v)dv {Vnm = *') , (7S)
C \ iV mJjfnn J
where Fs(p)dv is the net flux of radiation in the frequency interval
{v, V + dv) in the ^-direction.
1 ) The equation of transfer . — Consider a small cylinder of cross-
section da and length ds normal to da. Let 1^ be the intensity of the
radiation of frequency v on one face of the cylinder and in the s-c\i-
rection. Let the intensity emergent through the second face in the
same direction be ly + dly. The amount of radiant energy travers-
ing da in an interval of time dt and in directions confined to an ele-
ment of solid angle dco about the ^-direction is lydvdoidadt. Of this,
the amount of energy Kypds lydvdcadadt is absorbed by the cylinder.
Let jy be the coefficient of emission. The mass of the material
inside the cylinder is pdads; hence, the amount of radiant energy
emitted by this element of mass in the frequency interval {v, v + dv)
and in directions confined to the element of solid angle do) is
pdads jydo)dvdt . ( 7 )
Therefore, if the state is steady, we should have
dlydvd(j)dadt = pjydvd<jodadtds — pKylydvdcjdadtds ,
or
dfy _ . ^ .
pds
The foregoing equation is generally referred to as the “equation of
transfer.’' Of course, we have to consider ly as a function of posi-
tion and of direction; if it is necessary to refer to this explicitly, we
may write
ly = Iy{;x, y, z; I, m, n) , (70)
where the direction cosines (/, w, n) refer to the direction we are
considering. In a Cartesian system of co-ordinates we can write the
equation of transfer in the form
( d d d \
(77)
( 7 «)
RADIATION AND EQUILIBRIUM
199
In terms of the Einstein coefficients introduced in (/) and (^), we
can write the equation of transfer (78) in the form (cf. Eqs. [35]
and [42])
= Nn{Anm + — NmB,nnhv„mL„^ ,
or
^ . (82)
2. The thermodynamics of radiation. — We shall now investigate in
some detail the properties of radiation fields in systems adiabati-
cally inclosed. We shall first consider the case of a homogeneous
isotropic medium which, since we assume it to be adiabatically in-
closed, must be characterized by the same temperature T through-
out the medium. If we restrict ourselves to regions sufficiently dis-
tant from the walls of the inclosure, it is clear from considerations
of symmetry that in such regions the radiation field must be homo-
geneous and isotropic. In other words, the specific intensity of
j/-ra(liation must be independent of the position and the direction of
the ray. I'Vom the equation of transfer (78) it follows immediate-
ly that
jv = Kvif . (S3)
In other words, the ratio of the emission to the ahsor lotion coefficient for
the radialion of f requency v in the interior of a homogeneous isotropic
medium adiabatically inclosed is equal to the specific intensity of the
radiation for frequency v. This is one of KirchholT’s laws o[ radia-
tion.
If we express the (‘mission and the absorption coelTicients in terms
of the Einstein coellicients as we have done in equations (35) and
(42), we obtain from (83)
" 1 “ ~ ^ (S4)
or, solving for we have
j ^ nni
(85)
200
STUDY OF STELLAR STRUCTURE
which can also be written as
Kirchhoff ’s law in the form we have now derived is stated only
for those regions of the medium which are very far from the walls
of the inclosure, since it is only in these regions that we can derive
the homogeneity and the isotropy of the radiation field. It is, how-
ever, relatively simple to remove this restriction and to show that
ly has the value 7 V for all directions and all points arbitra' ily near
the walls of the inclosure. For, in an adiabatic inclosure, every pencil
of radiation must be characterized by the same value for h as the
pencil of radiation traveling in the opposite direction, since other-
wise there would be a unidirectional transport of energy. Hence, a
pencil of radiation emergent from an element of the surface on the
walls of the inclosure must be characterized by the same value for
ly as the pencil traveling in the opposite direction and coming from
the interior of the medium. An immediate consequence of this re-
sult is that the state of the radiation is the same on the surface of the
walls of the inclosure as in the interior. This result is also due to
Kirchhoff.
We have thus shown that the specific intensity, 7„, of radiation
of frequency v in an isotropic homogeneous medium adiabatically
inclosed depends only on the temperature and the nature of the me-
dium. We will now show, following Kirchhoff, that /„ does not also
depend on the nature of the medium.
For this purpose consider a small element of mass dm in the form
of an infinitesimal cylinder of cross-section da and height d$ normal
to d(T. Let p be the density of the material, so that dm = pdads.
Let the element dm be at the center of a hollow spherical reflector
of unit radius Which has, at the opposite ends of a diameter, two
small equal infinitesimal openings of area w; as the notation implies,
we assume that da is very small compared to co. Let the whole sys-
tem be adiabatically inclosed by an inclosure the inner surface of
which is ^^perfectly absorbing” while the outer surface is a “perfect
reflector.” We shall further suppose that the inclosure is completely
RADIATION AND EQUILIBRIUM
201
evacuated, so that a pencil of radiation is not weakened by absorp-
tion except when it strikes the element of mass dm, or the spherical
reflector (the outer surface of
which is also perfectly absorb-
ing), or the inner walls of the in-
closure itself. Finally, let the
whole system considered be at
temperature T (see Fig. 21).
Since the radiation field inside
the inclosure is isotropic, it follows
from our remarks in § i (/) that
the element of mass dm will radiate
energy uniformly in all directions.
Let be the emission coefficient. The energy radiated by dm in
unit time through each of the infinitesimal openings o) in the fre-
quency interval {v, v + dv) is
jyOidmdv = pjifoodadsdv . (87)
The energy emitted by dm in all the other directions is reflected at
the inner surface of the spherical mirror and, after repeated reflec-
tions, will again be incident on dm; it will thus be reabsorbed eventu-
ally by dm.
Now the walls of the inclosure radiate toward the interior only
since the outer surface is a perfect reflector. Part of the energy emit-
ted by the walls passes through the two openings on the outer sur-
face of the spherical inclosurc containing dm, strikes the element
dm, and is partially absorbed. The elements of surface of the walls,
2, and 2^, which arc accessible to the element dm have the areas
cos Ox
0 ) ;
cos Oj
( 88 )
where Ti and are the distances of dm from 2i and 2^, respective-
ly, and 0, and are the angles which the normals to the elements
2, and 2^ respectively make with the direction r which connects
the middle point of da and w.
202
STUDY OF STELLAR STRUCTURE
Now the total energy of y-radiation emitted by the element S,,
which is incident on d<r per unit time, is, according to equation (17),
( 8 ,)
where is the specific intensity of the y-radiation emergent from
Si in the direction making an angle 6i to its normal, and 0 is the
angle which the normal to da makes with the direction r. By (88)
we can write, instead of (89),
BI^^ cos 6 osdadv . (90)
The amount of this energy absorbed by the element dm is given by
BI^^ cos 6 o)d<Tdv Kpfi sec B ds KvpBl^^oidadsdv , (91)
In the same way, the amount of energy absorbed by dm in unit
time from the total radiant energy of v-radiation emitted by S^ and
incident on da is given by
Kv (jidadsdv , ( 92 )
where is the specific intensity of the ^/-radiation emitted by
Sa in a direction making an angle 0 a with its normal. Hence, the
total amount of energy absorbed by dm in unit time from the v-n\~
diation is given by
Kpp{B\,^^ + B^^^)o)dadsdp . ( 9 :^)
Now, since the system is in a steady state, the energy emitted by
the element through the two openings must be equal to (93). I"rom
(87) and (93) we have
= 2> . (94)
The foregoing equation remains unaltered if the walls of the inclosure
are deformed, thus varying the angles G, and Gj. It follows, then,
that
= B , ,
(95)
RADIATION AND EQUILIBRIUM
203
or the intensity of the radiation emergent from a black surface is inde-
pendent of the direction of the radiation. We can now write (94) as
KuJBit — Ji> . (9^)
If different black surfaces are taken for the inner surfaces of the
walls of the inclosure while the element dm is kept unchanged, it
follows that Bv remains constant. In other words, the intensity I^ of
the radiation emitted by a black surface is independent of its nature
and is a function only of the temperature. Finally, comparing (96)
with (83), we see that h = Bp, We have thus proved: The ratio
]p/kv of the emission to the absorption coefficient of any body in thermo-
dynamical equilibrium depends on the temperature only and is inde-
pendent of the nature of the body; further ^ ipf Kp is equal to the specific
intensity of the v-radiation emitted by a black surface. This is Kirch-
hoff’s law in its complete form.
Thus we have shown that Bp = >/ is a universal function of
temperature and frequency. About this function Bp thermody-
namics makes one important prediction. The energy density, w, of
radiation in an adiabatic inclosure at temperature T is, according
to equation (29),
and by Stefan’s law (proved in chap, ii) we have
Hence, if we denote by B the integrated black-body intensity, we
have
im = Ciwodu = ^ 7 '* . ( 99 )
Jo 4’r
For the inlejfratL'cl inlonsity, B, it is customary to write
« = <^ = 7 - (99O
x 4
a is called the 'Radiation constant,”
204
STUDY OF STELLAR STRUCTURE
We shall not go into the details of the derivation of the function
B,{T) at this place; the derivation is given in chapter x on the basis
of the quantum statistics. We note here that the quantum theory
predicts for By{T) the expression
^ . (loo)
where h is the Planck constant and k is the Boltzmann constant.
Equation (loo) expresses the well-known Planck law, and the ex-
pression for Bp{T) is often referred to as the “Planck function.”
Comparing (86) and (loo) , we see that in thermodynamical equi-
librium we should have
ly mBmn ___ i,, i^'p
Bnn. ~
(lOl)
Finally, we observe that Planck’s law enables us to evaluate the
radiation constant a and the Stefan-Boltzmann constant a in terms
of the fundamental constants h, c, and h. For
Uo,)
or, writing x = hv/kT, we have
Now,
nf'rs 2A {kTy x^dx
- F It) X
(103)
x>dx r“
+ e -^+ (104)
or, integrating term by term, we obtain for the integral, the series
+ + (ros)
Hence, we have
2ir^k^
RADIATION AND EQUILIBRIUM
20S
Comparing this with (99'), we obtain
3. Local thermodynamic equilibrium . — The thermodynamical the-
ory of radiation described in the previous section is valid only when
the system is adiabatically inclosed, and, as a result, when all parts
of the system are at the same temperature. Nevertheless, we often
encounter physical systems which, though they cannot be described
as being in rigorous thermodynamical equilibrium, may yet permit
the introduction of a temperature T to describe the local properties
of the system to a very high degree of accuracy. The interior of a
star, if in a steady and static state, is a case in point. For, even if
the temperature at the center of the sun, for instance, were 10® de-
grees, the mean temperature gradient would correspond to a change
of only 6 degrees in the temperature over a distance of 10^ cm.
This fact, coupled with a probably high value for the stellar ab-
sorption coelTicient, enables us to ascribe a temperature T at each
point P such that the properties of an element of mass in the neigh-
borhood of P are the same as if it were adiabatically inclosed in
an inclosurc at a temperature T. Under these circumstances we shall
say that the material in the neighborhood of the point P is in ‘‘local
thermodynamical equilibrium. ’’ In particular, if and > arc the
cocfiicients of absorption and emission of an element of mass, we
should have
> = k,,Bp[T) , (108)
where BXT) is the Planck function and T is the local temperature.
In using the foregoing equation, we have to remember thatj^ will
depend on the incident intensity of the radiation in the frequency
interval {v, v + dv) (cf. § i [/]). It is therefore more convenient to
use, instead of (108), the equivalent relations (Eq. [loi]) between
the Einstein coefficients :
^ ntn 2 ^
Bnm
^ mBmn
A nBnm
(109)
where T is the local temperature.
2o6
STUDY OF STELLAR STRUCTURE
Let us now examine the steady-state set up in a medium in a
static condition and which is in local thermodynamical equilibrium ;
this type of equilibrium was first studied by Schwarzschild.
Consider the equation of transfer in the form (cf. Eq. [82])
N nm NmB
mn hVnm ( N nB
-w — ; ; — v-nsz)'--- <■”>
Introduce the absorption coefficient K^m as defined in equation (42) :
We can write
or by (109)
Kv
nm
NJBmnhv nm
P
^ n J Anm N iJBnm
— Ann,hVn,n = ^ .
(ill)
(II 2 )
(II3)
Hence, we can write the equation of transfer in the form
dip o
— ^ = K, (i - . ( 114 )
pds ^ J ''nm V ‘
Suppressing the suffixes n and m and remembering that
= ^ -I)-'. (t^- 5)
we can re-write (114) as
M. p § /it f \
= k„B, - kJ, , (nO)
where
kI — K,(l — e-WAT) ^
It will now be clear why we did not take the equation of transfer
in the form
(118)
dL .
pds ~
RADIATION AND EQUILIBRIUM
207
and simply insert ioi jp the Kirchhoff expression (108); the reason
is that jp, which includes the induced emission, is in general not a
scalar but depends on the incident intensity But in the form (i 16)
we have allowed for the induced emission by reducing fc„ by the ap-
propriate factor [i — exp {—hv/kT)], The equation of transfer in
the form (116), with kI defined as in (117), is due to Rosseland.
4, The equation of radiative equilibrium and the solution of the
equation of transfer for the far interior , — We shall now solve (116)
under the circumstances applicable to the interior of a star. For this
purpose consider a medium which is in a static state and which ex-
tends (for all practical purposes) to infinity in all directions. Let us
further assume that the material is in local thermodynamical equilib-
rium. We shall suppose that a gram of material generates per unit
time an amount of energy €pdv by processes of an irreversible char-
acter. (Wc shall refer to €,dv as the ‘‘heat liberated,” including in this
term the net gain of heat per unit mass by an clement of mass by
“convection,”'^ “conduction,” and finally by the internal energy
converted into heat. Under the last item we include the [subatomic]
energy sources of a star.)
Now the total spontaneous emission per gram per unit time by an
element of mass at temperature T is
47 rK'B, .('/’) . (119)
The total absorption, less the total induced emission, is given by
(c-f. Eqs. [,53] and [47I)
(c'j/.f/co. (120)
Hence, the excess of emission over absorption, given by
<(■(/>’. - (121)
must, in a steady state, equal the heat liberated, tp. Hence, the con-
dition for a steady state is
«rf UK “ fp)do} ~ e,. . (122)
The foregoing equation is generally referred to as the “equation of
radiative equilibrium.”
4 This term is used loosely.
2o8
STUDY OF STELLAR STRUCTURE
We shall now write the equation of transfer (ii6) in a Cartesian
system of co-ordinates:
Multiply (123) by do^ and integrate over the complete sphere. By
(122) the right-hand side reduces to tpp. Hence, by equation (9),
which defines the flux components F*, Fy, and Fg, we have
or
dFxiy) , ^Fy(y) dF^jv) ^
dx dy dz ^ *
(124)
div Fp = €pp .
(125)
Equation (125) is simply a statement of the conservation of energy.
Now multiply the equation of transfer successively by Idcj, mdo),
and ndo), and integrate over the complete sphere. By the definitions
for the components of the stress tensor (Eq. [61]) we have
^ ^ + ^ = -T «') ' <"'■)
(„8)
dx dy dz c '
or
div A (129)
c
We shall now proceed to solve the equation of transfer. Let
.V, S
K^pds ; ds = Idx -f- mdy ndz , (130)
*■0* ^Oi So
the integral being taken from a fixed point (:ro, yo, Zo) and in the
direction {I, m, n). The solution of the equation of transfer can be
written as
Jp(^Xoj yo) Zqj Ij w, fi) ~ ^ Fi/( Tj ^6 dTv . (131)
RADIATION AND EQUILIBRIUM
209
The physical meaning of (131) is: The specific intensity at a given
point and in a given direction is simply the sum of the contributions
of the emission due to all the elements of mass behind (i.e., in the
direction negative to the one defined) the point under consideration,
after allowance has been made for the weakening of the separate
pencils of radiation from the different elements of mass, owing to
the appropriate amounts of intercepting material.
We assume that we can expand 5 „(— t„) as a Taylor series. We
shall retain only the first three terms in the Taylor expansion, and
we shall presently verify that this is sufllcient for a high degree of
accuracy. Hence, we write
jj / \ D f \ dBp I 1 a
‘Til) — Bv[q) Tv "T 2“^^
(132)
Inserting the foregoing in (13 1), we have
7,.(.v„, y,„ s„; I, m, n) = Bp{o) —
\dTp J TpP’O \ dT^ Jti,=o
(133)
Now
(I Bp _ dl^ _ _i_
d-Tp kIp ds KpP
DBp
<lx
+ VI
<lBp
<>v
+ n
OBp I
(134)
or
dBp
dTp
— grad. Bp .
Kf,p
(13s)
In the same way,
210
STUDY OF STELLAR STRUCTURE
Instead of (133) we can now write
Ip = Bp ^ grad, 5 ^ + 4 grad, ( 4 - grad, Bp] .
KpP ° kIp ° \KpP ° ]
(138)
Inserting the foregoing in the equation of radiative equilibrium, we
have
€pp = J 1 grads Bv — grad« grad, -Bv .
(139)
Since
and
= fmndo) = fnldcj = o
(140)
fl^dco = = fn^dti) = ^ ,
(141)
we have by (139), (134), (136), (140), and (141)
4 ir ^ ^
3 dx \/c'p dpX ) '
y. s,
(142)
From (138), (139), and (142) it follows that the ratio of the succes-
sive terms in (138) are of the order of magnitude
k:b.
(143)
where e is the total amount of heat liberated in all the frequencies;
B, the integrated Planck intensity; and k, a mean absorption coelli-
cient. Now in the interior of a star e ^ 100 ergs per gram per sec-
ond, K ^ 100 gm“* cm=* and T ^ 10® degrees. Hence, the ratio of
the successive terms in the expansion is of the order of magnitude
100
100 X - X io^-»
TT
^ 10”'^ .
(144)
Therefore for all practical purposes it would be sufficient to write
h = Bp
grad. Bp .
KyP
(us)
RADIATION AND EQUILIBRIUM
2 II
From the foregoing it follows that
= fu^,^ = ^B = ^T*= aT^,
pxxiy) = Pyy{v) = pxx{v) = pr{v) =
Pxyiy) = = p2x{v) = O . (149)
In other words, the stress tensor reduces to a simple hydrostatic
pressure. From (126), (148), and (149) (or more directly from [145])
we now have
, /,;(,) = _ 4^ ^ ; 1
Zk',p dx ?,K,p dy J
Fr = — ^ grad /L = — ^ grad pM .
7 ,K„P K„p
Finally, we have the exact relation (Eq. [125])
div F, = «rp . (152)
Equations (151) and (152) are the fundamental equations of the
jiroblcm.
We shall next consider the equation for the integrated flux. From
the first of the equations in (150) we have
.. . f
P'x{v)d» =
47 r r”i d/L ,
3P. jo k ’,' 'l-i' ^ ’
(15.0
I a/L , a?’
Z ar '^"aJ-
(154)
212
STUDY OF STELLAR STRUCTURE
We now define the coefficient of opacity, k, by
Equation (154) can now be written as
ZKpJo 9* 3«P 9x
(iSS)
(156)
where B is now the integrated Planck intensity. Similarly, if Fy and
F, are the integrated fluxes in the Y- and the Z-directions, we have
Thus,
4ir SB _ p _ dB
3 KP 9 y ’ ‘ 3 KP dz ’
F = grad B = — — grad Pr ,
3«P KP°
(157)
(158)
where pr is the radiation pressure Thus for the integrated
flux we have an equation of exactly the same form as (151), pro-
vided we average (k 0 ~' over the frequencies suitably. According
to (155), the coefficient of opacity, k, is a sort of harmonic mean of
k',. Explicitly,
I
K
ylQ2}ivfkT
_ 1)3 Ip
“ hv
(159)
where we have substituted the Planck function J 5 „ in (155). The
formula (159) for k is due to Rosseland, and for this reason k is also
called the “Rosseland mean absorption coefficient.”
Equation (152) in the integrated form can be written as
div F = €p , (ido)
where e is the total amount of heat energy liberated by unit mass in
unit time over the whole frequency interval {o ^ v ^ «»).
RADIATION AND EQUILIBRIUM
213
For a spherically symmetrical distribution of matter, equations
(151), (158), and (160) take the simpler forms
and
C dPr { v )
k',p dr ’
(161)
(162)
i « ■
(163)
where Fr{v) and Fr are the monochromatic and the integrated fluxes,
respectively, across elements of surface normal to the direction of
the radius vector r.
5. The equations of hydrostatic equilibrium.- 'Consider a thin
cylinder of unit cross-section and height ds in the direction (Z, m, n).
By equations (75), (log), and (117) the mechanical force exerted
by radiation on the cylinder in the frcciuency interval {v, v H- dv)
in the s-direclion is
K[,pF^{v)du
c
(164)
By (151) this can be written as
— grad pM • ds . (105)
Hence, the mechanical force exerted by radiation in all the fre-
quencies is obtained by integrating (165) over all the frequencies.
By Stefan’s law we then have
— grad (Jf/T'O • ds .
On the other hand, if py is the gas pressure, then this material
pressure gradient would exert a further mechanical force on the
cylinder considered of amount
(167)
— grad pg • ds .
214
STUDY OF STELLAR STRUCTURE
Let V be the gravitational potential. For hydrostatic equilibrium
we should have
grad + p,) = -p grad V. (168)
For a spherically symmetrical distribution of matter (cf. Eq. [7], iii)
equation (168) is equivalent to
^ iPo + ^^ 7 ’“) =
GM{r)
P
(169)
The other equations of equilibrium are equations (158) and (160);
for a spherically symmetrical distribution of matter the appropriate
equations are (162) and (163). When these equations are used, the
quantity L{r), which is the net amount of energy crossing a spherical
surface of radius r, is generally introduced instead of Fr. Then,
47rr^
(170)
In terms of L{r) equations (162) and (163) take the forms
and
47rcr*
(171)
dL(r)
~^ = 4 -^rUp. ( 172 )
Equations (i6g), (171), and (172) are the equations of equilibrium
for a star in radiative equilibrium.
BIBLIOGRAPHICAL NOTES
§ I.— The most complete accounts of the theory of radiation are contained
in —
1. M. Planck, Wdrmestrahlung, sth ed., Leipzig, 1923; English translation
(of the 2d German ed.) by M. Masius, Planck’s Heat Radiation, Philadelphia;
Blackiston, 1914.
2. H. A. Lorentz, Lectures on Theorciicd Physics^ 2, 209-275, 1927.
From the point of view of astrophysical applications the most valuable ac-
counts are contained in —
3. E. A. Milne, Handb. d. Astrophys., 3, Part I, 1930.
RADIATION AND EQUILIBRIUM 215
4. S. Rosseland, Asirophysik auf atomthearetischer Grufidlage^ Berlin: Spring-
er, 1931.
§ 2. — For the thermodynamics of radiation see Planck (i) and Lorentz (2).
Also —
5. P. Drude, The Theory of Optics, Part III, chap, ii; English translation by
C. R. Mann and R. A. Millikan, New York: Longmans, 1922.
The most rigorous proof of Kirchhoff’s law is due to —
6. D. Hilbert, Physik, Zs., 13, 1056, 1912.
7. D. Hilbert, Physik. Zs., 14, 592, 1913.
See also —
8. H. Straubel, Physik. Zs., 4, 114, 1903.
Hilbert’s papers are very illuminating, and a careful study of them is well
worth while.
§ 3. — The conception of local thermodynamical equilibrium is due to —
9. K. ScHWARZSCHiLD, Gottingcr NochricMcn, p. 41, 1906.
The derivation of Eq. (116) with as defined in Eq. (i 17) is due to S. Rosse-
land:
10. S. Rosseland, Ilandb. dcr Astrophys., 3, Part I, 443-457, 1930.
§§ 4 and 5. — 'rhe conception of radiative equilibrium is also due to Schwarz-
schild (9), but the analysis in the form given in § 4 is due to Rosseland (4); see
also E. A. Milne (3).
Equations (162) and (163), which are fundamental in the theory of radiative
equilibrium, are due to
11. A. S. Eddington, M.N., 77, 16, 1916. See also ibid., 77, 596, 1917;
79, 22, 1918.
Eddington did not, however, consider the problem of radiative transfer
in the separate frequences.
'Fhe physically correct derivation is due to Rosseland (10). Also —
12. S. Rosseland, M.N., 84, 525, 1924.
CHA.PTER VI
GASEOUS STARS
In the last chapter we showed that the equation of hydrostatic
equilibrium of a spherically symmetrical distribution of matter in
radiative equilibrium is
+
(l)
where pg is the gas pressure, pr{= is the radiation pres-
sure, and the rest of the symbols have their usual meanings. If the
stellar material is a perfect gas, we can write
P — pa + pr = pT + \aT ^ .
( 2 )
The radiative temperature gradient is determined by (cf. Eqs. [171]
and [172], v)
^ o Q)
and
dr 4Trcr^ '
dL(r) = 4irr^pedr .
(4)
In discussing the structure of model gaseous stars in radiative equi-
librium, we have to exercise considerable care, inasmuch as we do
not, as yet, know the exact dependence of e on p and 7 \
In this chapter we shall attempt a first discussion of the equations
of equilibrium for a gaseous star in radiative equilibrium. The prob-
lem of the ‘‘stability” of the radiative temperature gradient will
also be considered.
We shall begin our discussion by proving a few integral theorems
on the radiative equilibrium of a gaseous star.
I. Integral theorems on the radiative equilibrium of a star. — Divid-
ing equation (3) by equation (i), we have
dpr ^ Kljr)
dP 4TrcGM{r) ‘
216
GASEOUS STARS
217
We introduce the quantity rj, defined by
V =
Ljr)
Mjr)
M
( 6 )
where L is the luminosity of the star. As defined in this manner, ri
is the ratio of the average rate of liberation of (“the heat”) energy
e(r) interior to the point r, to the corresponding average e for the
whole star:
( 7 )
r, -
From (7) it follows that
viR) = I ; Vc = ^,
in an obvious notation. Inserting (6) in (5), we have
(ipr _ L
dP ^'kcGM
KTI
( 8 )
(9)
Integrating the foregoing equation from r = r to r = R and using
the boundary condition = o at r = R, we have
Following Stromgren, we now define the average value icij(r) by
iivir) ^ f
(In writing equation [11] we have used the boundary condition
that P = o at r = R.) Hence, we can re-write (10) as
Pr =
Let
Pr = ii- 0)P ; A = PP •
(12)
(13)
2i8
STUDY OF STELLAR STRUCTURE
Then by (12) and (13)
Knir) = 2 \ L
(14)
We have thus proved the following theorem due to Stromgren.
Theorem i. — The ratio of the radiation pressure to the total pres-
sure at a point inside a star in radiative equilibrium is proportional
to the average value of ktj for the regions exterior to the point r, the aver-
age being taken with respect to dP, where P is the total pressure.
As a particular case of (14) we have
^ ^ 4x^Gilf(i - ff.)
KTJ
where Icrj now defines the average value for the whole star. Equa-
tion (15), which is an exact equation, is a formula for the luminosity,
L, of a star in terms of its mass, M, and an average value of kt). We
shall refer to equations (14) and (15) as the luminosity formulae.
Sometimes it is useful to have an equation similar to (15) but which
involves an average value of (i — jS) instead of the value of (i — fi)
at the center. A formula of this kind can be obtained as follows :
Write equation (10) in the form
Multiply both sides of the equation by P’dP and integrate from o
to Pc We then have
? +
Integrating the right-hand side by parts, we obtain
(18)
GASEOUS STARS
219
Define the following averages :
(l $)n = 1
^)» ~ F^fa )
( 19 )
''lC 1 J(l{P'‘) .
10
In terms of these averages equation (18) can now be written as
( 7 ^),+. == ^
which is the required formula. If g = o, we have, as a special case
of the foregoing,
L =
2K’qi — Krji
Theorem 2. — IfKfj{r) decreases oultvard from the center in a star
in radiative equilibrium, then (i — j8) must also decrease outward from
the center.
This is an immediate consequence of equation (14).
The following theorems, 3 and 4, are due to Chandrasekhar.
Theorem 3.- 7 m a wholly gaseous configuration in radiative equi^
librium in which the mean density p(r) inside r docs not increase out-
ward, %ve have
^ y- , (23)
where jS* satisjics the quartic equation
M = Y-^V3 uzZ!
\r} \nJI) a / 3*4
Proof: Since we have assumed that the mean density does not in-
crease outward, we can apply Theorem 7 of chapter iii, according
to which
I - / 3 , ^ I - , (25)
220
STUDY OF STELLAR STRUCTURE
where |8* satisfies equation (24) and is determined uniquely by the
mass M. Combining (25) with the luminosity formula (15), we ob-
tain
^ ^ - /3c) ^ 47rcGikr(i - /3*)
KT) ^
(26)
which is the required result.
Theorem 4. — In a wholly gaseous conjiguraiion in which the mean
density p(r) inside r and the rate of liberation of energy e do not increase
outward, we have
^ (27)
where k is a mean opacity coefficient defined by
and where the equality sign in (27) is possible only when e is a constant
throughout the configuration.
Proof: This is an immediate consequence of Theorem 3 above.
For, if e decreases outward, then ri(= £(r)/e) must also decrease
outward, and consequently the minimum value of ?j is unity. Hence,
^ ^ . (-*<))
where, according to (ii), the opacity k is to be defined as in (28)
above. The equality sign in (29) is possible only when 77 = c-on-
stant = I, that is, when e(r) = e = e = constant throughout llie
configuration. Combining (23) and (29), we have the reciuired re-
sult.
We shall apply equation (27) to certain practical cases of interest.
Numerically, equation (27) reduces to
K ^ 1.318 X - 18*), (30)
where Lq refers to the luminosity of the sun.
For the sun, on the assumption that /x„ = i, the solution of the
GASEOUS STARS
221
quartic equation (25) (obtained by interpolating among the figures
given in Table 2, iii) yields i — jS* = 0.030. Hence, by (30)
< 391 cm^ • (31)
For Capella, on the other hand, Jkf = 4.18O and L = 12OL0; on
the assumption that /Zc = i, i — jS* = 0.22. According to equation
(30), we now have
iccapciia < 100 gm~' cm" . (32)
We shall sec in chapter vii that the stellar opacity coefficient plays
a fundamental role in the further development of the theory; it is
therefore satisfactory to have the upper limit (30) proved under ex-
tremely general circumstances. It should be noticed that the meth-
od of averaging weights the central regions of the configuration very
heavily, and hence the upper limit (30) is essentially an upper limit
to the opacity at the center of the configuration. The inequality
(30) can be interpreted in the following manner:
If, for a star of given mass M and luminosity L, ic should be great-
er than the limit set by (30), then, either the density or the rate of
generation of energy, €, or both, must increase outward in some
finite regions of the interior of a star.
We can prove a somewhat less sharp ineciuality for k at all points
in the star, 'J"he. following theorem is due to Eddington.
Tuicorf.m 5. For a i^iiLscous slur in radialivc cquilibrmm in which
Ihc densilyj Icmperalnrc, and the rale of liberation of energy do not
increase outward, we have
^ ^ircCtM
(33)
at all poinls inside the configuration.
This theorem is an immediate consequence of ccpiation (9). For,
if p(r) and T{r) do not increase outward, dpg will be positive (or
zero), for i)ositive increments in p and 7 ’, and must always be less
than dP. Hence dpr/dP must be less than unity. By (9) we should
therefore have
^ttcGM
(34)
222
STUDY OF STELLAR STRUCTURE
If, further, the rate of generation of energy, €, does not increase out-
ward, then, as we have already pointed out, tj ^ and hence,
by (34)
j^ircGM * ^ ^
(35)
which proves the theorem. It should be noticed that the upper lim-
its for K and K differ only by the additional factor (i - /3*) in the
expression for the latter. For stars of small mass this factor can be
very small (e.g., for the sun i — j8* = 0.03), and thus the upper
limit for k is physically of greater interest.
Finally, we shall derive a very useful alternative form of equa-
tion (9). Combining equations (9) and (14), we have
dP »
(36)
which can also be written in the form
dpr _ K7J dP
Pr ~ Kvir) P ’
(37)
again, since p, = \aT\
dT 1 K7] dP
T ~ 4 Kiiir) P •
2. Stability conditions for radiative equilibrium . — If a spherically
symmetrical distribution of matter is in hydrostatic equilibrium,
and if further radiative equilibrium obtains, then the radiative tc'in-
perature gradient is determined by
dT I Kv dP
T 4 P ’
where
Lv))
p.±,T + iaT>.
(40)
We shall now consider the stability of the radiative gradient : To
examine this, suppose an element of mass hn, originally at tempera-
ture T, density p, and pressure P, suffers a sudden increase of tem-
perature of amount hT > o. Then this element exerts a pressure of
a definite amount, 5 P > o, on its surroundings and expands and be-
GASEOUS STARS
223
comes less dense than its immediate neighborhood. The element hm
will then experience a force tending to displace it into regions of
lower density. During such a movement the element continues to
expand, the temperature altering in the meantime.
We shall now make the following assumptions : (a) that at each
instant of time the element bm expands (or contracts) to such an
extent that the pressure exerted on the element by the surrounding
material is the same as that which the element exerts on the sur-
rounding material; (6) that the process of expansion (or contraction)
takes place adiabatically ; (c) that the viscous forces restricting the
movement of the element bm can be neglected. We shall first ex-
amine the consequences of these assumptions.
By our second assumption, since the expansion of bm takes place
adiabatically, we should have, according to equations (124.) and
(134) of chapter ii.
where
/dT\ 1\ - I dP
\T r, P ’
(4 - .^|3)(t - i)
+ 3(7 — i)(i — ^)( 4 - + P) ’
(41)
(42)
for Ihe rate of change of the temperature of the element bm as it
moves outward into the regions of lower density, expanding in the
meantime.
C'omparing (39) and (41), we see that the temperature of 5 m, as
it moves outward, alters at a rate different from that of its imme-
diate surroundings because, according to our first assumption, the
pressure P alters in the same way for both 5 m and the surroundings.
Let us now suppose that
r, — 1 . I Ki;
r, ^ 4 «j(r) •
(43)
'I'hen it follows that the element bm, after moving outward for a cer-
tain distance, will find itself at the same temperature, pressure, and
density as its surroundings at that point ; consequently the original
disturbance dies out.
In the same way, if the element bm originally suffers a decrease of
temperature of amount 5 T, then it will become denser than its im-
STUDY OF STEULAR STRUCTURE
224
mediate neighborhood and consequently will sink to regions of high-
er density. If we make the three assumptions as before, then the
adiabatic compression it experiences as it sinks to regions of higher
density increases the temperature of dm at a rate greater than the
local temperature gradient. Again, it will soon find itself at a point
where dm and its neighborhood at that point have the same density,
pressure, and temperature. Thus in either case, i.e., either for a
positive or a negative increment dT of an element dm, the disturb-
ance dies out if equation (43) is true. In this sense the radiative
equilibrium is stable if the radiative gradient is less than the correspond-
ing adiabatic gradient. This result is due to Schwarzschild.
On the other hand, if
r, — 1 I Kii
r, 4 Kiiir)
(44)
and if the element dm suffers an increase of temperature, then, as
before, it will move outward to regions of lower density; now, how-
ever, the temperature of dm will decrease less rapidly than that of
its surroundings, and hence it is always at a temperature higher
than its surroundings. In the same way, if the element dm suffers a
decrease of temperature, it will sink to regions of higher density;
but the adiabatic compression it experiences (according to our as-
sumption [6]) will not ever be sufficient to raise the temperature
of dm to that of its surroundings (if equation [44] remains true);
consequently, it always remains cooler than its surroundings. Hence,
we have proved on the basis of the assumptions that the radiative
equilibrium is stable or unstable according as
4
r. - 1
r.
> or <
<0?
If^(r) ■
(45)
Table s gives the values of 4(r„ - i)/r, for different values of
(i - &)•
TABLK 5
—
1“^
0
O.I
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 .(J
4 (ri-r)
1 \
1.6
1-304
1. 1 77
I .108
1.065
1.039
1.022
1 .010
[ .004
1 .000
GASEOUS STARS
225
3. The equations of equilibrium when the radiaiim gradient is un-
stable , — Suppose that we have initially a situation in which the ra-
diative gradient obtains but one which exceeds the adiabatic gra-
dient. By our discussion in the previous section, the radiative gra-
dient is unstable, in the sense that a slight alteration of the local
temperature will give rise to a system of ascending and descending
currents which will have the effect of reducing the existing tempera-
ture gradient. Eventually a steady state must be set up, though it is
not a priori clear what the nature of that steady state will be. Under
the circumstances, the general picture which is adopted is the fol-
lowing one.
We suppose that masses of gas are continually being detached
from the surrounding matter and that they move bodily through a
certain distance before they are reabsorbed into the main mass of the
material. Alternatively, the situation can be described by saying
that ‘‘eddies” are continually being formed which travel, on the
average, a distance I with a certain mean speed u before being re-
absorbed into the main mass of the material. The quantities / and u
thus defined are referred to as the “mean free path” of the eddies and
the “mean speed of turbulent motion,” respectively. We further
suppose that we can define a certain mean temperature T at each
point to describe the local properties.
Consider the transfer of heat energy across an element of surface,
S, at r, which is large compared with the cross-sections of individual
eddies. The eddies which are absorbed into the main mass of ma-
terial at r will have been formed, on the average, at points distant I
from r.
The eddy which is formed at (r — Z) will have a temperature
When the eddy appears at r, and before it is reabsorbed, it will have
a temperature
where {dT/dr)'^ is the rate at which the temperature of an eddy alters
during its motion. If we assume that during its motion an eddy
226
STUDY OF STELLAR STRUCTURE
expands or contracts adiabatically to such an extent that the pres-
sure exerted by it on its surroundings is equal to that exerted on it
by its surroundings, then
/ dlogT \* ^ r, - I dlogP
\ dr ) Tj dr '
At r the eddies are reabsorbed into the main mass of the surround-
ing material at constant pressure. Hence, the total energy, Q, cross-
ing the surface, S, and expressed in terms of unit area, is
where Cp is the specific heat at constant pressure of the matter and
radiation (cf. Eq. [146], ii). The foregoing expression can be writ-
ten as
where the eddy conductivity, a, is defined by
a = pul = ^\ pul dZ .
^J(S)
It should be mentioned here that a will itself depend upon the de-
gree of instability as specified by {{dT/dr)* - {dT/dr)}. Indeed,
we should expect that with increasing instability the turbulent mo-
tions will become more violent; this would, in turn, lead to larger
values of u and, hence, of <t. In general, the magnitude of the eddy
velocities will be determined by the balance of energy which be-
comes available to the eddies from the mean internal energy, and
the energy lost by the eddies through viscous dissipation.
Further, th e mean value of the rate of transfer of momentum
across S is pw’ per unit area. Also, the transport of turbulent energy
is measured by \pu{u^ + v‘ ->r w‘)^
' «, V, and w are the components of the eddy velocity with respect to a fixed frame
of reference. Further, it is assumed that u is in the direction of r.
GASEOUS STARS
227
When writing down the equations of equilibrium, it should be
remembered that
dpr __ KpFr
dr c ’
(47)
where pr is radiation pressure and Fr is the actual net flux of radiant
energy. If c has the same meaning as in equation (4), we then have
4rr^F.,
■"’X
— 47rr^
epr^dr
idr
+ ^ pu{u^ -f + V)^)
■ (48)
The equation of hydrostatic equilibrium can now be written as
dr
(pr + + pwO
GM(r)
”17“ P
(49)
Equations (47), (48), and (49) arc quite general. A more detailed
discussion is needed to make these equations more explicit. The case
of vanishing radiation pressure has been investigated by Cowling,*
whose results we shall quote :
(a) The transporter turbulent energy, ^puUF + in (48),
and the turbulent pressure, p//*, in (49) can be neglected in compari-
son with the other terms occurring in the respective equations (6).
The temperature gradient as delincd by (47) and (48) dilTers from
the adiabatic gradient (dT/dr)* only by a very small amount. The
temperature gradient set up will therefore be only very slightly
superadiabatic.
The foregoing simplifications seem to arise mainly from the cir-
cumstance that the internal energy of a gram of the material is so
very large compared to the energy loss, €, due to subatomic proc-
esses. Consequently, even very slight mass motions are sufl'icient to
reduce a superadiabatic gradient to a stable one which dilTers from
the adiabatic gradient only by an insignificant amount. Thus, on
the basis of the Bicrmann-Cowling analysis we can conclude that
* Essentially equivalent results were given by L. Hiermann.
228
STUDY OF STELLAR STRUCTURE
when the radiative gradient becomes unstable, we have an adiabatic gra-
dient set up with the relations
p az py oz .
The case when the radiation pressure is comparable to the gas
pressure has not yet been fully investigated.
Finally, it should be pointed out that, if the convection currents
become violent, we may have to introduce entirely new considera-
tions. In particular, the three fundamental assumptions of § 2 need
not necessarily be valid. For, if the ‘‘inertia of motions” (to use
Kelvin’s phrase) is large, then the elements of ascending and de-
scending masses will experience viscous friction, which may result
in the communication of probably quite appreciable amounts of
heat to the eddies during their motions.
4. The standard model, — In the Introduction we pointed out the
fact that an attack on the problem of stellar structure is made pos-
sible at the present time only on the basis of certain assumed laws
concerning the rate of generation of energy, e, or the energy-source
distribution, as defined by 77. A model which was first introduced by
Eddington and which has played an important role in subsequent
developments is the so-called “standard model.” This is defined as
one in which ktj is a constant throughout a given configuration.
From the luminosity formula (Eq. [14])
we infer that (i — jS) is a constant throughout the configuration.
Since we can write (cf. Eq. [87], iii)
it follows that for the standard model we have the relation
P = Kp4/3 ;
GASEOUS STARS
229
where, if we further assume that /x is a constant, isT is a constant.
The equilibrium configurations are therefore polytropes of index
» = 3, and the general theory of chapter iv can be applied. In par-
ticular, the Lane-Emden function 63 completely determines the struc-
ture of the configurations. We have
P = F,ei -, p = pM : T = ta , (S3)
where Pc, po, and Tc are the values of the variables at the center.
From §§ 6 and 7 of chapter iv we have the following results.
a) The mass relation— 'Ey equation (70) of chapter iv and equa-
tion (52) above, we have
M =
-47r
/AV 3 I - P
Xp-H) a \
(S4)
in other words, ^ is determined uniquely by M and satisfies a quartic
equation. Equation (54) was first derived independently by Bialob-
jesky and Eddington. It is of interest to compare (54) with the
quartic equation for (3* (Eq. [24]), which gives the minimum value of
^c at the center of a star of given M, in which the density does not
increase outward. By comparison we find that the ratio of the nu-
merical coellicicnts in (54) and (24) is given by
— ^ X 2 0182 = 3.296 . (ss)
it"'- \ (/?/«=«. tt'
Table 6 gives the values of M for dilTerent values of i — j3.
TAHI.K ()
(i —/i) FOR TiiF Standard Modkl
i-H
0.5
(Sy
50 . 86
4 - 45 ^>
o.()
S7.04
7.020
0.7
167.1 ()
1 2 . 5O
0.8
0
0
20.10
o.()
iro.s-y
31 5 <)
1.0
CD
230
STUDY OF STELLAR STRUCTURE
b) The ratio of the mean to the central density . — ^According to equa-
tion (78), chapter iv, and Table 4, we have
Pc - —
3 ^
p = S4.i8p .
(56)
t={i
c) The central pressure . — ^According to equations (80) and (81) of
chapter iv and Table 4 we have
P. =
GM‘
GM^
-c def\ V
("TF
R- - “■°5 R> •
or, numerically,
Pc= 1.242 X IO'7
cm”
d) The central temperature . — ^We have for the present case
Uj PcT, = = /SPc .
From (56) and (57)
To =
\-^Ti
GM
R
or
r - r,
A -0.8543 — -^.
Numerically, the foregoing is found to be
To = 19.72 X /3 m X io« degrees.
(S7)
(58)
(59)
(60)
(61)
(62)
e) The potential energy . — By equation (90) of chapter iv we have
GASEOUS STARS
231
/) The mean temperature . — If T is the mean temperature de-
fined by
MT = f TdMir) ,
then
Mf = = ^j^PdV ,
(64)
(6s)
or, since jS is a constant,
Mf = -- ^ n ,
(66)
or by (63)
- _ I GM
2 k R '
(67)
Numerically, the foregoing is found to be
f = II. S4i8m X lo" degrees .
(68)
g) The internal and the total enerf’y. 'Phe internal energy con-
sists of two parts: the contribution by the gas and the contribution
by the radiation. A slight modification of Ritter’s relation (Eq. [55],
iii) yields
IJ - P -il
“ 3(7 - i) ’
(69)
where 7 is the ratio of siiecific heats of the gas. 'Phe internal energy,
gra.i, due to the radiant energy, is
1
II
r*
II
(70)
or for the standard model
UrM = -(l - m .
(71)
Hence, the total internal energy is
U = + Urart = - | I + “
, (72)
232 STUDY OF STELLAR STRUCTURE
where Q is given by (63). The total energy, E, is seen to be
E=u + a= a. (73)3
h) Nutnerial applications . — ^As an example of the application of
the foregoing formulae, we shall calculate the values of P,., p,„ and
Tc for three typical stars — the sun, Sirius A, and Capella A. (More
extensive tables are given in Appendix III.)
TABLE 7
M/O
R/RQ
l/lq
Pc
pc
To
Sun
I.OO
2.34
4.18
1 .00
1.78
iS -9
1 .00
38.9
120
0.003
0.016
0.04s
1 .2X10*7
6.8X10*6
3.4Xio*‘»
76.5
31 ■?
0.080
20X 106
26 X 106
SX106
Sirius A
Capella A
* H has been assumed to be equal to unity in all cases (cf. chap. vii).
5. The luminosity forimda for the standard model.- lly the lu-
minosity formula we have, quite generally, that
_ ^ tcGM{i — /3r)
KTj
(74)
For the standard model, (i — 0 ) and kt] are constants; and conse-
quently we can write
1 — fic = 1 — ; KH = KcVr. ,
and the luminosity formula can be written as
_ 4 ircGAf(i — g)
KcVc
( 7 S)
(7())'
3 If jS is not constant, the general relations are
and
( 73 ')
PdV.
(7.5')
^ reason for writing the equation in this way will be clear from the discussion
in § 7 .
GASEOUS STARS
233
For the coefhcient of opacity we assume a law of the form
K = KopT-^-‘ , (77)
where «« is a numerical constant which may depend upon /*. We can
quite generally re-write (77) as
® ^ rp — jt
* = r •
3 « I - P
(78)
Substituting for (according to [78]) in (76) and using equation
(60) for the central temperature Tc, we get
L =
^xcGM
On the other hand, from the quartic equation (54) we have
(80)
4’+* \ k ) 3 [..Wj] ' k„t;,: R"
(81)
Eliminating (i - / 3 )" from (79) and (80), we have
L
For the case s = 2, equation (81) is numerically found to be
^ . 793 X .0- (I)’ ■ (ff)" ' . (87)
Equation (81), then, is the mass-luminosity-radius relation for the
standard model. It is, of course, clear that, since on this model the
stars form a homologous family, must be the same for all stars;
it is a pure number.
6. llomolof’ous Iransformalions. -In the previous section the {L,
M, R) relation was derived for the standard model. We have now
to examine the question as to how general the results based on the
234
STUDY OF STELLAR STRUCTURE
standard model can be expected to be.® This problem has a twofold
character, (i) How general is the form of the relation (8i)? (2) How
can we apply the standard-model distributions of density and tem-
perature to configurations in which ktj is not accurately constant
but shows only slight variations from constancy?
Regarding the form of (81), we shall prove, following Stromgren,
that /or a star in radiative equilibrium^ in which the radiation pressure
is negligible throughout the configuration,
T
L — constant (83)
Kq Jx
if the rate of generation of energy, €, and the coefficient of opacity follow
the laws
€ = €op“r*' ; K — /CopT“ 3 -s , (84)
where a, v, and s are arbitrary. The constant in {83) depends only on
the exponents a, v, and s.
Proof: The equations of equilibrium can be written as :
dP _ GM{r)
dr r‘
(8s)
(86)
(87)
3 ^ 2 T ' —
j — l^oP JL '
dr ^ac
(88)
s In an investigation {Ap. 87, 535, 1938) completed since the writing of the mono-
graph, an important minimal characteristic of the standard model has been proved. It
can be shown that/or gaseous stars in equilibrium in which p and (i — p) do not increase
outward the minimum value 0/ (i — is the constant value of (i — fi) ascribed to a
standard-model configuration of the same mass. For stars in radiative equilibrium the
condition that (i — should not increase outward is equivalent to ^(r) not increas-
ing outward.
GASEOUS STARS
23s
In writing equation (87) we have neglected radiation pressure, ac-
cording to our assumption.
The system of equations (8 5)- (88) has to be solved with the bound-
ary conditions
M{r) = M , p = o, T = o at r = R (89)
and
M(r) = o at r = o . (90)
These provide four boundary conditions; and since the system of
equations (8s)-(88) is equivalent to a single diflferential equation of
the fourth order, it follows that there is just exactly one solution
which will satisfy the foregoing boundary conditions. We shall now
show that from such a solution we can construct another solution
such that it will describe another configuration with a different
Rj and p; wc shall see, in fact, that the transformation required to
go over from one set of values, M, i?, and p, to another set, Af 1, Ri,
and At,, is the successive application of three elementary homologous
transformations. To show this, we proceed as follows:
Let the physical variables, after a general homologous transforma-
tion has been applied, be denoted by attaching a siilTix “i”. For a
general homologous transformation we should have
Ti = , M, = y”sjLt ,
Pi = , 7\ = ,
M{ri)i = , (ic*,€„), = y”7(/fo€„) ,
Pt = ,
where fh, . . . . , fiy are, for the present, arbitrary constants and y
is the transformation constant. The exponents (w,, .... , W7) should
satisfy certain relations, namely, those which are necessary for the
continued validity of equations (85) (8tS) in the sullixed variables.
Substituting (c;i) in ((S5), we find that we should have
yn.-.i = , ( 92 )
or
712 — w, = 7/3 + — 2«, .
(93)
236 STUDY OF STELLAR STRUCTURE
In the same way, equations (86), (87), and (88) yield:
= 2«x + «4 , (94)
»2 = M4+«6-Ms, (95)
«6 — «1 = M7 + (a + 3)^4 — (6 + 5 — v)n6 + fit . (96)
We thus have four equations between the seven unknowns. Hence,
we should be able to express any four of «’s in terms of the other
three. We shall choose fit, n^, and «s as the independent quantities.
Solving for Wj, n^, and n^ in terms of n,, %, and «s, we find:
= -4«I + 2»3 ,
= — 3 «l + «3 > (98)
»6 = — + «3 + «s , (gg)
«7 = —{s — v — it^fit + (4 + 5 - y _ -I- (7 + 4. _
If we choose fit = i, % = o, and = o, we have a homologous trans-
formation in which a star of a given mass ilf and molecular weight ix
is expanded or contracted. In the same way, the choice n, = o,
% = I, and »s = o corresponds to an alteration of M while R aiul
g are kept fixed. Finally, the choice fit = o, = o, and = t
corresponds to an alteration of fi while R and Af are kept unchanged.
These three elementary homologous transformations are schemati-
cally represented by
ft = Vnr
= y-R^P
Af(fi)i = M{r)
Pt =
Ml = P
T’l = yTT
(/Coeo)i = y /“+■’-"(«„«„)
Rt = y^R
ft = r
P^ = yhP
Mir,), = yj,Mir)
P. = yuP
Mi = M
= yifT
r, = r,
I\ = P,
Mir,), = Mir) ,
P‘ = P ,
M. = y,jx ,
P^ = y;p,
(«»«o)i = M, = ■■■(«„«.,) ,
M, = yi^M n, = ^
(101)
We have now to consider how the luminosity is changed by a homolo
gous transformation. Since
r
= 47r I
pfdr ,
(102)
GASEOUS STARS
237
we have, according to our law (84) for e,
KoL = 47 r/Co€o^ p°‘'^^T‘^dr .
(103)
Hence, by a general homologous transformation,
where
KoL alters to
(KoL)i = 3 ;" 7 + 3 n,+(a+i)«^+Wi 6 ^^ ^
(104)
or by (97), (98), (99), and (loo)
{KoL)i = 3 /-*««i+(S+«)» 3 +( 7 +«)ws(xoL) .
(los)
In other words,
L = constant — — .
(106)
It is clear that the constant in (io6) can depend only upon the ex-
ponents .9, Vj and a.
We have thus proved the invariance of the form of the luminosity
formula for stars in radiative ecpiilibrium. If, however, the law of
energy generation is such that it leads to a suniciently strong con-
centration of the energy sources toward the center, then we should
reach a stage when
Kv ^ ^ r.- — I
^ I’;. ■
(107)
In other words, going inward* toward the interior of a star, the
radiative gradient will become unstable at some delinite point
r = fi, (say). For stars with negligible radiation we have from
(107) that
KT}
(r = ri). (107O
For r < n, Kv/lcvir) > 1.6. Now the right-hand side of (107') is a
pure number, while the (luantity on the left-hand side is homology
invariant. Hence, l/te frarlion, ci = ri/R, of the radius at which the
^ We shall see in chap, viii that the radiative griulient is stable in the outer parts
incluflint? tlii! stellar envelope. '
238
STUDY OF STELLAR STRUCTURE
instability of the radiative gradient sets in is the same for all stars with
vanishing radiation pressure. The fraction q depends only on the ex-
ponents s, a, and v, which occur in the laws for k and €.
According to the discussion in § 3, the material interior to
n = qR will be in convective equilibrium. Since the radiation pres-
sure is assumed to be negligible, in the convective core we should
have
where pi and p* refer to the pressure and the density at the ^'inter-
face,” i.e., at r* = qR, Equation (108) is clearly homology invari-
ant. Hence, the structure of the convective core is also homology in-
variant. We have thus proved the invariance of the form of the
luminosity formula (106), quite generally.
We have stated and proved Stromgren’s theorem for strictly van-
ishing radiation pressure. It is, however, clear that if /3 is very nearly
unity, the variation in jS can be properly neglected^ and a mean
value chosen. The result is equivalent to defining a new “molecular
weight” jjlP instead of jjl; in other respects, the method of argument
remains as before. Hence, we have, more generally than (106),
L = constant i) , (109)
which is identical in form with the (L, M, R) relation derived for
the standard model. The present restriction that the radiation pres-
sure is negligible means that we should restrict ourselves to stars of
small mass (cf. Theorem 7, iii). We shall see in chapter vii that the
majority of stars for which we have observational material concern-
ing L, M, and R fall into the class of stars with “negligible radiation
pressure.” Thus, the use of the (L, Af, R) relation derived on the
basis of the standard model can be largely justified- especially as
we now sec that the same form for the relation results for a wide
class of stellar models.
Again, since the stars form a homologous family under the reslric-
7 This is not the same thing as neglecting the variation of i —
GASEOUS STARS
239
tions of Stromgren’s theorem, we can apply Theorem 13 of chapter
iii, according to which
I — ^ I — go / MiV
(MoiSo )4 \Mo) ^
(no)
where and po refer to the values at the corresponding points in
two stars of mass Mi and Mq. Hence, we should have, as a particu-
lar case of (no),
I — Pc = constant M^CixPrY . (no')
In other words, (i ~ P,) should satisfy a certain quartic equation —
the constant in (no') will of course be different from that in the
quartic equation for the standard model.
We thus see the complete parallelism between the standard model
and these more general models.
7. Perturbation theory for varying ktj.- -The nature of the problem
presented can be described in the following way.
We first assume that kt) is constant; this leads to a perfectly defi-
nite distribution of density and temperature. Now a physical the-
ory, on the other hand, may be expected to si)ecify the precise de-
pendence of the opacity and the rate of generation of energy on the
density p and the temperature T. From the march of p and T, de-
rived on the basis of the constancy of kt], we can calculate k and ri at
each point and form the product ktj; a test of the consistency of the
model is that the product X77, determined in this way, should be
reasonably constant. If this is so, the question arises as to how we
can apply the results based on the hypothesis of ktj being constant
to cases where ktj shows slight variations from constancy. The an-
swer to this question can be given only on the basis of a perturbation
theory, which we shall proceed to outline. The following analysis
is a modified version of the theory which was first developed by
Strdmgren.
Now, the luminosity formula predicts that
I - ^ =
L
4TrcGM
Kiiir) .
(ill)
240
STUDY OF STELLAR STRUCTURE
We can re-write (in) in the form
KTJ
(112)
If we make definite assumptions concerning the dependences of
K and 6 on p and T, we can evaluate the march of the quantity K^(r)
inside the star on the basis of the standard model. Equation (112)
will now specify the variations in (1 — 0 ) as determined by the
luminosity formula.* We suppose that the variation of (i — 0 ) thus
predicted is small and that we can write
-
0 i
1/3
J
(l + ^)p4/3 ,
(1I3)
where we can regard ^ as a small quantity of the first order. It will
be noticed that in writing (113) we have assumed that the varia-
tions of both I — jS and jS as determined by (112) are small. This
implies that for values of i •— jS,, near unity, the permissible range
of variation for kt] is much narrower than when i — /S,. is small;
for example, a variation of ktj by 10 per cent will be permissible for
I — jSc = o.i, while a variation even by this amount should be ex-
cluded for I — jSc = 0.8 or 0.9, if the standard model is to be re-
garded as a reasonable first approximation.
We assume that as introduced, is a known function of r; rf/ will
be simply related to l^{r)/l^. We write (113) as
P = A'(i + , (i 14)
where
In the equation of hydrostatic equilibrium,
1 d dP\
P dr \p dr ) 4 ^ 'P >
(iiO)
® Vo/ as determined by the local values of density and temperature.
GASEOUS STARS
241
set (cf. Eq. [10], iv)
p = X 03 ; P = , (117)
Equation (ii6) reduces to
In the foregoing equation ^ is to be regarded as a known function
of 5 . At ^ = o, ^ must satisfy the boundary condition,
^ = 0; ^ = 0 (? = o). (120)
Further, ^ = o at the boundary. Also, ^ is a small quantity of the
iirst order and is to be regarded as arbitrary otherwise.
In (119) write (Kelvin’s transformation)
We have
.V
£
4 (lx
I
(T
(i +
-O’ .
(121)
(122)
To solve the foregoing equation, we shall assume that we can write
0 = 6 + X , (123)
where 6 satisfies the Lane-Emden equation
and where x is a small quantity of the first order. Equation (122)
can now be written as (if quantities of the second order are neglected)
, <1
(lx
,{x^ (lx (lx ^ ■' (lx
-fl’- 3 ®'x, (125)
or
(I'd (l-d
dx^
dx‘
5 ^ ^ ^
"^4 dx i/r
^ = -
- 3®'x ; (126)
242
STUDY OF STELLAR STRUCTURE
or, using (124), we have
We can also write the foregoing as
S - («■£)■ (»*)
Reverting to the variable in we have
^ ^ = “3^x + - ^4 1 . (129)
which is a linear differential equation for x- Equation (129) has to
be solved subject to the boundary conditions
dx
^ = o at J = o ;
X = 4 ' = o at 1 = ^. , (130)
where ^ is the boundary of the Lane-Emden function 83; we have
chosen 63 for 0 in (123). The boundary conditions (130) are clearly
necessary and are, further, sufficient to determine x uniquely. Wc
have thus solved the formal problem of obtaining a second approxi-
mation.
The mass relation is easily found to be
M = - 4 ira 3 X ^ . (131)
or, using (120), (123), (130), and the boundary conditions that 6^
satisfies (Eq. [67], iv), we have
(132)
which, on substituting for a (Eq. [i i8]), is seen to be a quartic equa-
tion for I - ft. Hence for a given M, equation (132) determines
(i ft). To use the luminosity formula
jr _ 4‘^cGM(i — ft)
(1.33)
GASEOUS STARS
243
we have to determine kti by
j rPc
~ To jo
or by (117)
^ ° + ^)Q^] > (13s)
or by (123)
If If
^ ^ + 4 ^X) • (136)
Hence, we can re-write (133) in the form
L - ~ ^•'> , (.37)
where
5. - (Ti^ X""' i
a) First approximation . — It is clear from our analysis that a first
approximation can be obtained by using Eddington’s quartic equa-
tion to determine (i — jS,.) and by evaluating %. in the luminosity
formula (137) by (cf. Eq. [138])
(139)
If we assume for k a law of the form already used in § 5, equations
(77) and (78), then, for the standard model distribution of density
and temperature
- = . (140)
Kr.
Hence, by (t3q)
• (141)
The quantity rj,., determined by (139), is a homology-invariant con-
stant and is the same for all stars. We thus see that the luminosity-
244
STUDY OF STELLAR STRUCTURE
mass-radius relation derived in § 5 (Eq. [81]) can be used as it stands
if rjc in equations (81) and (82) is replaced by tJc-'
L
43+s
[oW 3]4^" Ko^c -R"
(142)
As an illustration of the method, we shall consider the case where
€ varies as some power of temperature and the opacity varies accord-
ing to (140), with s = Table 8 shows the run of rjd~^ with 0*^ as
argument.
TABLE 8
«3
€ = Constant
goc ya
6 « 7 '*
I
I.OO
1.70
2.57
4.71
0-9
1. 01
1 .69
2 . 53
4.40
0.8
1.02
1 .69
2.48
4.0K
0.7
1.04
1 .69
2.40
3-«5
0.6
1.06
1.68
2.34
3 . 65
0-5
1,09
1.68
2 . 27
3.40
0.4
1 . 12
1.67
2 . 20
3 - 14
0-3
1 . 16
1. 6s
2.13
2.87
0.2
1 . 22
1.67
2.06
2 • 55
0. 1
1.33
1. 71
I .96
2 . 24
An examination of Table 8 shows that for the cases e = const ant
and € ~ r the first approximation can be safely used at an>’ rate,
for stars with negligible radiation pressure. For the case € = con-
stant, the value of can be evaluated directly from (141):
Vc (« = constant) = 4j^ = « = i .
If KTj were accurately constant, then for this model in the luminosity
formula (76) (or [142]) we should strictly have 77, = i . 'Finis our
approximation probably introduces an error of jo per cent in the
luminosity formula. (A more detailed investigation of the model
[k cc p7^3.s^ ^ constant] given in chapter ix confirms the present
conclusion.)
^ For c oc r, the standard model is a very good approximation wit li
Vc 1.68 in the luminosity formula. The model ceases to be good,
GASEOUS STARS
245
in the first approximation, when e varies with a higher power of T,
For these cases a second approximation will be necessary. However,
in using the formula (142), it is customary to adopt (following Ed-
dington) a value of 97^ = 2.5; this probably corresponds to a rather
high value for the concentration of the energy sources toward the
centre.”
b) Second approximation. To obtain a second approximation, it
is necessary to solve equation (129) for x- We shall write equation
(129) in the form
, 2 dx
^ + = + . (144)
where
The boundary conditions arc (cf. Eq. [130])
dx
^ = O at J=o; x = o at { = . (146)
These boundary conditions arc at diJTerent points; hence, if we wish
to solve for X directly, it would be necessary to adopt a method of
trial and error. This can, however, be avoided by first solving the
corresponding homogeneous equation
d^X I 2 dx
tie ^ di (^ 47 )
and by then obtaining x by quadratures. 'J'his is the method of
the variation of the parameters.
^ Let Xi and x= be any two linearly independent solutions of (147).
Then the solution of (144) can be written as
X = ^UOxi + , (148)
where A (f) and B(^) arc, for the present, two unknown functions,
which, however, arc restricted to satisfy the relation
dA , dB
° • (H 9 )
246 STUDY OF STELLAR STRUCTURE
By (148) and (149) we have
and
dk
( 150 )
^ ^ ^ + R , d/t dx. , dB dx. , ,
d!^ df- ^ ^ ■*" If If + :« IT • . (151)
df* df df ^ dk di
By substituting (150) and (151) in (144), and remembering that
Xi fl-nd X2 satisfy the corresponding homogeneous equation, we find
i^^,dBdx 2 _ „
dk rff W ~ ■
(152)
We can now solve for dA/d^ and dB/d^ from (149) and (152). We
find
■ dk
Xi
dXi
dxi
n(f)
and
d^
^
d^ dx.
dxi
n(f)
(153)
(154)
Integrating the foregoing equations, we have
A =
and
■
J, «
n(i)d$ + c,
(iss)
^n(i)d| + c3, (156)
d^
where c. and c, are two integration constants, which have to be
determined from the boundary conditions (146). Since x has to van-
ish at f „ we immediately have that
Cl Xi + C, Xi = o
(f = i) (iS7)
GASEOUS STARS
247
The other boundary condition yields (cf. Eq. [150])
In choosing the two linearly independent solutions Xi and X2 of
(147)1 we can arrange that one of them (say Xi) is such that
dxi
di
o at f = o .
(159)
If Xi has been chosen in this manner, then from (157) and (158)
we have
Hence, we have finally
In this way the problem can be formally solved. For applications
to practical cases we shall need Xi and Xj- Once Xi and Xj are known,
then for any given ll(^) two quadratures are sunicient to determine
the appropriate solution x for (144).
STUDY OF STELLAR STRUCTURE
2|8
BIBLIOGRAPHICAL NOTES
§ I. — !• B. Stromgren, Handb. d. Astrophys,, 7, 159, 1936, where Theorem i
is proved.
2. S. Chandrasekhar, Ap, 7., 86, 78, 1937 (Theorems 2, 3, and 4).
3. A. S. Eddington, Zs.f. Phys., 7, 351, 1921 (Theorem 5).
§§ 2 and 3. — The stability criterion for the existence of radiative equilibrium
was first considered by —
4* K. Schwarz SCHILD, Goitinger Ncchrichtetif 1906, p. 41. The discussion in
the text follows —
5. L. Biermann, Zs.f. Ap., 5, 117, 1932. Also A.N., 257, 270, 1935.
6. H. SiEDENTOPF, A.N.J 244, 273, 1932, where a condition equivalent to
equation (45) is given.
7. S. Chandrasekhar, Proc. Nat. Acad. Sci. (Washington), 23, 572, 1937.
8. T. G. Cowling, M.N.i 94, 768, 1934.
§ 4- — The standard model was first considered by Eddington — (ref. 3, above).
Also,
9. A. S. Eddington, M.N., 77, 596, 1917, and The Int-ernal Consiituiion of
the StarSj chap, vi, Cambridge, England, 1926.
10. I. Bialobjesky, Bull. Acad. Sci. Cracoviey May, 1913 (p. 64).
§ 5. — See ref. 9, above. Also,
11. E. A. Milne, Handb. d. Aslrophys.y 3, Part I, particularly pp. 204-222.
§ 6. — See ref. i, pp. 168-170. Also,
12. B. Stromgren, Erg. exakt. Naturwiss., 16, 465, 1937.
§ 7.— The discussion in this section, which is based on an unpublished in-
vestigation by B. Stromgren, differs in principle from similar discussions in
references (9) and (ii). Further reference may be made to —
13. R. Emden, Thcrmodynamik dcr Himmclskbrpcr (‘‘Sonderausgabe aus der
Encyklopadie der Mathematischen Wissenschaften”).
CHAPTER VII
STROMGREN’S INTERPRETATION OF THE
HERTZSPRUNG-RUSSELL DIAGRAM
In the last chapter the mass-luminosity-radius relation for gase-
ous stars in radiative equilibrium was derived. The relation in ques-
tion (due essentially to Eddington) was first derived on the basis of
the standard model; but, as we have seen, the same form for the
relation results for a wide class of stellar models if the radiation pres-
sure is not very appreciable. Further, by a perturbation method we
have seen how the luminosity formula may be applied to cases where
K 7 } is variable. For most practical purposes it is sufficient to restrict
ourselves to the first approximation considered in the last chap-
ter (§ 7).
In this chapter we shall be concerned mainly with concrete appli-
cations of the luminosity formula to the available observational ma-
terial regarding the masses, luminosities, and radii of the stars.
I. llte statement of the problem^ On the first approximation con-
sidered in chapter vi, § 7, we have
where
__ 4%cCiM{\ — ft.)
— — ,
(l)
KrVr
( 2 )
Further, (i — is determined once M and the mean molecular
weight are known, as the solution of lOddington’s quartic equation.
Now we shall show in § 5 that the physical theory of the stellar
opacit}^ coefficient k leads to a formula of the type
_p_
t '
(3)
240
250
STUDY OF STELLAR STRUCTURE
where «« is a constant depending on the molecular weight, and t,
called the “guillotine factor” (Eddington), is a slowly varying func-
tion of p and T. We can write equation (3) as (Eq. [78], v)
/ 3 A I - /S
Inserting the foregoing in the luminosity formula (i), wc have
^ 4ircGM 3 (i -
apff j3c ‘ ’
where ijc can now be expressed in the form
=
( 6 )
which can also be written as
ijo = I . (7)
In the foregoing equation Hs a certain harmonic mean value of I de
fined by
i_Xl
ndei
( 8 )
Now the integral on the right-hand side of (7) is the value of if
the guillotine factor were unity. We shall accordingly deline ^„(t ) by
By (7), then.
(9)
(to)
From (5) and (lo) wc have
J _ ^ircGM (i — PcY 3 ^r/
{k„/1)Ui) Pc a y.11 ^
THE HERTZSPRUNG-RUSSELL DIAGRAM
251
Proceeding exactly as in chaper vi, § 5, we find (cf. Eq. [81], vi)
L =
/OTV-S £C fg-S
\ * / 3 [oWa]"'
I MSS
or, numerically (cf. Eq. [82], vi),
I MS'S
(12)
(13)
where L, M, and R are expressed in solar units. According to our
remarks in § 7 of chapter vi, we shall adopt ^,.(1) = 2.5. Inserting
this value in (13), we finally have
i MS'S
D = 7. 17 X io" 4 - ^ . (14)
Now if we know the mass and the radius of a star and if, further,
we assume a value for /i, then the foregoing formula enables us to
calculate L. In general, the value of L so calculated may not agree
with the observed value. We should, however, be able to adjust /x in
such a way that the observed and the predicted values of L agree.
In other words, a knowledge of Z, M, and R should enable us to de-
termine the mean molecular weight of a star, or, what is equivalent
to it, the mean chemical composition of the stellar material. This
is precisely our present problem. The solution consists essentially
in {a) determining the appropriate ju for stellar material of a specilied
chemical composition and at a prescribed density and temperature,
ib) determining the dependence of on the chemical composition,
and {c) determining the appropriate value of I for individual stars.
Once these questions have been settled, the determination of /x is
immediate. We can then compute the value of fj. for a number of
stars for which values of Z, M, and R arc available. Our final prob-
lem is to examine it these computed values of /x enable us to give a
general interpretation of the characteristic features summarized in
the Hertzsprung-Russell diagram.
We shall consider, following Strdmgren, these questions in the
order stated. It is necessary, however, as a preliminary to the whole
discussion, to consider a fundamental theorem due to Vogt and
Russell.
252
STUDY OF STELLAR STRUCTURE
2 . The Vogt-Russell theorem . — ^The theorem in question states that
if the pressure^ P, the opacity^ k, and the rate of generation of energy^ e,
are functions of the local values of p, T, and the chemical composition
only, then the structure of a star is uniquely determined by the mass and
the chemical composition.
To prove this theorem we shall first consider the case of a gaseous
star in radiative equilibrium. For this case the equations of equilib-
rium can be written in the form
GM{r)
r^
F
^ 47rr®
dP = —gpdr
dM{r) = 47rr*p(fr
■I ;
^ II ;
diyr*) = -—pdr
C
dL(r) = ^irr^epdr
III .
The foregoing system of equations can, in principle, be solved as fol-
lows: We choose a definite value for r and prescribe an arbitrary set
of values for the variables P, P, g, and Fr. From P and T we can
calculate the ‘‘local’’ values for p, k, and e; to deduce these values,
we require a knowledge of the chemical composition or its equivalent,
the mean molecular weight. The second set of equations above then
enables us to compute dP and dT for an increment dr of r. In the
same way the third pair of equations enables us to compute dM{r)
and dL{r). Thus we have a set of values for the variables P, T, g,
and Fr for r + dr. We can therefore continue the solution for a fur-
ther increment of r. In this way we can integrate the solution both
inward, toward the center, and outward, toward the boundary. For
a solution to be physically possible the following boundary condi-
tions must be satisfied :
and
M{r) = o at r = o
p — > o , P — ► o simultaneously ,
(15)
(16)
or, more exactly, p — > o and T To {o, definite limit), but this is a
refinement hardly ever necessary. We thus see that there are three
relations between the four values initially adopted for P, T, g, and
Fr, respectively. Hence, we are left with only one disposable con-
stant, Since at the end of the integrations we should be able to find
THE HERTZSPRUNG-RUSSELL DIAGRAM
253
the total mass M of the configuration, it follows that an assumed
fixed chemical composition can lead us only to a one-parametric se-
quence of configurations; the parameter can clearly be chosen to be
M. In other words, given M and we should, in principle, be able
to calculate the other two observable characteristics of the star,
namely, L and R — of course, on the basis of an assumed chemical
composition.
We have thus proved the Vogt-Russell theorem for gaseous stars
in radiative equilibrium. From our method of argument, however,
it is clear that the theorem is valid quite generally, i.e., also when
a part of the stellar interior is in convective equilibrium (cf. chap, vi,
§§ 2 and 3) ; for no new parameters are introduced. The theorem es-
sentially arises from the fact that the equations of equilibrium are
equivalent to one differential equation of the fourth order, while a
solution, to be physically possible, has to satisfy three boundary
conditions. Thus we have proved the general validity of the Vogt-
Russell theorem.
It is necessary, however, to point out that there are conceivable
physical circumstances under which the Vogt-Russell theorem will
not be valid. Thus, e need not, in general, depend upon the local
values of p and T; this would be the case if the origin of stellar
energy were due to physical processes occurring at nearly equilibrium
rates, e.g., nuclear transmutations occurring at approximately eciui-
librium rates but slightly more in one direction than in the other.
'J'here are, however, good reasons why such cases can be excluded;
we shall return to this question in the last chapter. Meanwhile, we
shall accept the validity of the Vogt-Russell theorem. The applica-
tion of the theorem we have in view is this: Docs the use of the lu-
minosity formula to determine the chemical composition of the stars
allow us cither to confirm or to deny the validity of the Vogt-Russell
theorem for stars in nature? We shall see that the answer to this
question is largely in the affirmative.
In the next two sections we shall consider the question of the mean
molecular weight and the theory of the stellar opacity coefficient.
Unfortunately, we cannot start from first principles in our discussion
of these two quantities, as we have done so far in the treatment of
STUDY OF STELLAR STRUCTURE
the other problems. The importance of these quantities in the pres-
ent connection, however, makes a treatment — even if only a partial
one — essential. In the following two sections we shall assume a gen-
eral familiarity with methods of statistical mechanics and of the
quantum theory.
3. The mean molecular weight oj highly ionized stellar material , —
The first problem is to determine the appropriate molecular weight
that has to be used in the equation of state adopted:
This problem, as we shall see, is essentially one of determining the
number of particles per unit volume. For we have also
p„ = NkT . (18)
Suppose we have a mixture of elements and that an element of atom-
ic number Z occurs with an abundance factor Xz — in other words,
I gram of the material contains xz grams of the element. Let us
suppose, further, that each atom of the element contributes, on the
average, free particles per unit atomic weight, i.c., if A is the
atomic weight (that of hydrogen being taken as unity), then each
atom contributes free particles. We then have
N = jj:i:xznz. (iq)
where the summation is to be extended over all the elements. Ky
(18) and (19) we have
pg = jj {^x:znz)pT . ( 20 )
Comparing the foregoing with (17), we find
The determination of fx involves, therefore, the specification of the
state of ionization of the stellar material at a prescribed density and
temperature.
THE HERTZSPRUNG-RUSSELL DIAGRAM
2SS
a) First approximation. — We shall first give an elementary first
approximation. Let us suppose that the conditions are such that
the ionization is complete — that is, an element of atomic number Z
and atomic weight A gives rise to Z + i particles. Then
It is well known that, except for the lightest elements (hydrogen
and helium), the ratio defined as in (22), is approximately i : 2.
Hence, if we assume that in i gram of the stellar material there are
A" grams of hydrogen, Y grams of helium, and (i — A — Y) grams
of the “heavy” elements, then we can write
Hi = 2 ; Ha = f ; nz = h • (23)
The expression for ju then becomes
_ I
^ ~ - -V - ]') ’
or
2
^ I + _^X + o. 51' ■
(24)
(25)
If the helium content can be neglected, we shall denote the abun-
dance of hydrogen by A’,,; then (i — A',,) is the abundance of the
heavy elements. In this case
The general expression for the numlxT A,, of free electrons is
N, = ^ V (nz.I - i) . (27)
In the present apjiroximation (Mcj. [22]) we have
- JL
'' “ II ^ ' A" •
N.
(28)
2s6 study of stellar structure
If a mixture of hydrogen, helium, and the heavier elements is con-
sidered, we have, according to the abundances leading to (24),
Y. = |[X-h|F + |(i -X- F)], (29)
or
AT. = i (30)
h) Second approximation , — In order to determine fx more accu-
rately, it is necessary to calculate nz more accurately, with allow-
ance for the state of ionization. Stromgren has developed the fol-
lowing elegant method of doing this.
In this method an approximation is made in which the differences
between the states of the first and the second if-electrons, or be-
tween the states of any of the i-electrons, are ignored. The differ-
ences between the states of L- (or M-) electrons belonging to normal
or excited configurations are also ignored. Any atomic configuration
is then specified sufficiently by the numbers (nx, n^, . . . . ),
which give, respectively, the number of L-, M-, etc., electrons
bound to an atomic nucleus of charge Ze. The energy of such a state
may be taken to be
-riKX^P - (^^i)
where the x’s 3,rc constants representing the mean ionization po-
tentials of the various shells. Since, as we shall see presently, at
most common temperatures and densities in stellar interiors the
ionization is generally very far advanced, it is clear that, consistent
with our present scheme of approximation, it is suflicicnt to use for
the x’s the expressions derived for hydrogen-like atoms. If the nu-
clear charge is Ze, then by Bohr's theory we have
(Z) _
where n stands for the principal quantum number (n = i for the
A-electrons, w = 2 for the i-electrons, etc.,), is the mass of the
electron, and the other symbols have their usual significance.
THE HERTZSPRUNG-RUSSELL DIAGRAM
257
The statistical weight for the configuration (n^, n^f, ••••)?
for short (n), is given by
q{K, nK)q(L, nL)q(M, n^) • - • - , (33)
where, according to Pauli’s exclusion principle, it is easy to verify
that
^(A, HA') ~ ’ q^^^ i hm) “ ^ (34)
where the C’s are the binomial cocfllcicnts.
We now consider the equilibrium between the single configura-
tions (n), free electrons, and the corresponding bare nuclei, denoted
by (o, o, . . . . ) or (o). From statistical mechanics (cf. R. H. Fow-
ler, Statistical Mechanics^ 2d ed., Cambridge, 1936) we have
■■■■
Nin)
_ [G(r)]"A-+'’z,+"<w+ • • ■ • X ■■■■
g{K, nr^qiL, hm) ....
where N{o) and N(n) are the number of atomic configurations in
the states (o) and (n) and N,. is the number of free electrons per
unit volume and where 6 X 7 ") is defined by
G{T)
^ {nrvhkTyti
//•’
( 3 fi)
Equation (35) can also be expressed in the form
N{n)
N(o)
' ' ■ X [v(^', nsML, n,MM, n„) . . . . ]
X • • • • )/fiT .
The evaluation of the total number of bound electrons by this meth-
od can be sufficiently illustrated by the calculation of which
gives the average number of bound electrons with princij^al quan-
tum number 2 around a nucleus of atomic number Z.
Let
G{T)
y •
(38)
258 STUDY OF STELLAR STRUCTURE
Then, by (34) and (37) we have
nF =
8
^TiCrf
o
8
^sCr/
o
8y(i + y)i _ 8y
(i + y)“ I + y ■
(39)
The other shells give similar contributions. Quite generally, we sec
that the number of bound electrons, ThP, of principal quantum
number n, around a nucleus of atomic number Z, is given by
211^
(40)
Since in an unionized atom there are 2«’ bound electrons with i)rin-
cipal quantum number n, equation (40) corresponds to a number
of free electrons per atomic nucleus and arising from the ionization
of the M-shell:
2 »' — =
2n‘
I +
G{r)
( 4 «)
Finally, the number of free particles per nucleus of char):>;e Zc is
+ y-
2 Tr
I +
iV..
G{r)
Ml
jkT
(4-’)
where the summation is extended over all the relevant «’s. In jirae-
tice it would suffice to consider only K-, L-, and M-clcctrons. ( If it
should be necessary to consider higher orbits, then factors negli'cted
here, such as “excluded volumes,” should be taken into account.)
By definition we have
+ 2-
2 ?^
I +
No
6X7’)
JZ)
/kT
(43)
THE HERTZSPRUNG-RUSSELL DIAGRAM
259
If we have a mixture of elements with definite abundance factors,
then the molecular weight is, according to (21),
where
_ I _ I
n = '^xznz •
(44)
( 45 )
Table 9, due to Stromgren, illustrates the calculation of n for the
so-called “Russell mixture,” in which the elements 0 , (Na -1- Mg),
Si, {K + C(i), and Fe are assumed to occur by weight in the ratio
TABIJC 9
1
Total
Number
Nil mher
Number of
Element
n
of Bound
Electrons
of Eree
Electrons
Free Par-
ticles per
"z
Nucleus
0
1
0 <3
6 d
2
0.24
7.76
iS . 70
0 . 54 «
Cl
d
Na, Mg. . . .
1
5 6
6 6
0-3
11.7
12.7
•53
.132
o. i.Sj
Si
0 . 24!
■-»
O.II
0.5
>3 5
>4 5
• 5 *>
.032
0. 17J
K, Ca
1
1 4.6]
o-
1 .0
iS. l
10. 1
.4.S
• 030
P'c
I
d - d d
30
23.0
24 . 0
0.43
0.054
T = 10" (lrf;nTs; lof;
1 (^rn 1
1 ’ iV, 1
5 n
0.52
^ n
[ . ()2
8 : 4 : I : I : 2. We thus see that, given the temperature T and
the quantity Gi'n/N,., we can calculate n and fi. 'fable 10, also
due to Stromgren, gives the values of n« calculated for the Russell
26o
STUDY OF STELLAR STRUCTURE
mixture for different values of T and G{T)/^e- The reciprocals
of the values of tabulated in Table lo give the mean molecular
weight, M, for the Russell mixture. This is not generally far from 2.
TABLE 10*
FOR Russell Mixture
Log T
3
4
s
6
7
8
Q
10
64
0.46
0.40
0.50
0.51
O . S2
6.6
0.48
.51
•52
•53
■53
■ 54
6.8
0.4S
• 51
•53
•53
■53
■ 54
■54
7.0
0.44
• SO
•52
■53
■54
■54
■54
■54
7-2
.46
.51
•53
•54
.54
■54
■54
■54
7-4
.47
• SI
•53
.54
.54
■54
•54
■54
7-6
0.47
0.51
0.53
0.54
0-54
0.54
0.54
0 . 54
For a Russell mixture completely ionized, n^^=o.S4
* We shall use ‘Xog” to denote logarithms to the base 10 and ‘Mog” to denote natural logarithms.
On the other hand, if i gram of the stellar material contains .V
grams of hydrogen, F grams of helium, and (i — A' — F) grams of
the Russell mixture, then
or
n = 2 nzXz = 2X + f F + n,i{i - A' - )') , (40)
I
2X + f F + Hb(i - A - 10 •
(47)
If the helium content can be neglected, then F = o. A' = A',., and
we have
_ I
“ 2Z„ + nn(i - X„) ■’
or, solving for Xo, we obtain
( 4 «)
A-„
(49)
Equation (49) will give the hydrogen content after the value of ju
has been found from the mass, luminosity, and radius of a star.
THE HERTZSPRUNG-RUSSELL DIAGRAM
261
Finally, to determine /x for a given density p and temperature T,
it is necessary to know Ne. To determine Ne it is sufficient, in the
first instance, to use the result of the first approximation, namely,
equation (30) :
f + ^) • (so)
If necessary, it would be a simple matter to use a method of reitera-
tion. Accurately, i.c., in our present scheme of approximation, we
have, according to (41),
Z » I -h
2n^
(si)
G{T)
where the summation is extended over all the elements and all
the relevant principal quantum numbers, and where it may be re-
called that
Gcn =
(52)
c) The accurate dclcrniinalion of p. To determine /x more accu-
rately than by Strdmgrcn’s method, we must allow for differences
between the dilTercnt electrons in the same shell, for “excluded vol-
umes,'’ and also for electrostatic corrections. Such calculations have
been made, for certain special cases, by Fowler and (iuggenheim.
We shall not go into these refinements here, but reference may be
made to Fowler’s Statistical Mechanics.
4. The stellar opacity coefficient. The main contribution to the
opacity for the radiation in stellar interiors arises from photoelectric
ionizations of the electrons bound to the nuclei of the highly ionized
atoms. If the ionization potential for a particular state of the atom
considered is x, then radiation of frequency p ^ where
= X , (53)
can ionize the atom photoelectrically. In addition to these bound-
free transitions, there is still another kind, which can be described
as “free-free” transitions and which also contribute to the opacity.
262
STUDY OF STELLAR STRUCTURE
These free-free transitions correspond to free electrons which ab-
sorb quanta of a definite frequency while in the attractive field of
an atomic nucleus.
Now, since the atoms in the interior of a star will be highly ionized,
it is sufficient to consider for the probability of these bound-free and
free-free transitions those computed for hydrogen-like atoms, i.e.,
for assumed Coulomb fields around charged nuclei. According to
the theory of this phenomenon, as developed by Kramers, Gaunt,
and others, we have the following results.
a) Bound-free transitions . — If an atomic nucleus of charge Ze is
considered with an electron in the state of principal quantum num-
ber w, then the atomic absorption coefficient Ooiv] Z; n) is given by
( 54 )
(55)
if
Z:n).
•s Xn
" ^ = T
In the expression for Z; w), the quantity g{y\n), called the
''Gaunt factor,” is a factor which depends on n and v, and which,
for the values of n and v of importance in contributing to stellar
opacity, is very near unity. Table it, due to Stromgren, gives the
TABLE 11
v/v^
»= I
«=2
w = 4
I .0
0,80
0.89
0.94
0.98
I .00
0.99
0.88
0.94
0.97
1.02
1.04
1. 05
0.9
I .0
1.5
2.0
3.0
4.0
5.0
values of giy, n) for some values of n, with argument v/vn. As
vjvn-^ o; but, as will be shown in the subsequent discus-
sion, only such values of g arc of significance as arc very near those
at the series head. Hence, we can, in practice, regard g as independ-
ent of V and take as its value some constant value g. We shall re-
turn to this question later.
THE HERTZSPRUNG-RUSSELL DIAGRAM
263
b) Free-free transitions . — If we consider an atomic nucleus of
charge Ze, then the rate of absorption of energy ao{v; v), from radia-
tion of frequency v and of unit intensity by electrons of velocity v
and unit mean density, is
ao{v; v; Z)
47rZ^g^*
3 V 3 hcm\vH
(S6)
There is a Gaunt factor for (56) as well, but we shall take it equal
to unity (the free-free transitions do not in any case contribute ap-
preciably to the stellar opacity).
We shall now calculate the Rosseland mean coefficient of opacity
as a function of density and temperature:
Let us first consider the case of a single element of atomic num-
ber Z and atomic weight A . By (56) the contribution to the absorp-
tion coefficient per nucleus, expressed per gram of the material,
which arises from free-free transitions and which is due to electrons
with velocities in the range v, v -f dv, is
OiXv :
7 \n
N„(ii ) ,
( 57 )
where N is the number of electrons ])er unit volume in the speci-
fied velocity range. By Maxwell’s law of the distribution of veloci-
ties we have
N ,dv = 47rA,. 2 k’l ^
The absorption coefficient kIP ( °o, «»), due to the atoms under con-
sideration and arising from free-free transitions, is given by
JX)
(^
X f ' iiu{v; V\Z)
Mt
( 59 )
or by (56) and (5(S)
fy-.. . T 1 67r-Z- c^m,Ne
or, lifter integration,
{X)( ^ * T()7r=^" <;'■
4^1(00, ») = —
N„ 1
AH ^-5/3 hc{ 2 Trm,y^‘
(61)
364
If we put
STUDY OF STELLAR STRUCTURE
hv _
u,
(62)
we can re-write (61) in the form
AZ)
where
(...). 22,
V > / m3 ’
(63)
i6t“Z’ Ne
(64)
3-^/3 c{ 2 irm^^^^ (^r )3 s ■
Let us now consider the bound-free transitions. By equation (40),
the average number of electrons per atomic nucleus in the state of
principal quantum number n is
-(Z)
Hn —
2 n^
1 + ^
iV e
(65)
Hence, the contribution to the absorption coefficient which is due
to the electrons in the state n, expressed per gram of the material, is
where Vn is defined by
= X
By (54) and (66) we have
AH
(66)
v(a)
—
'^2
(67)
' yl// 3V3 Ch'" "" ’
( 68 )
where, according to the remarks on page 262, the quantity g{v\ n),
which occurs in (54), has been replaced by a constant g. Wc now
re-write (68) in the form
( 00 ^ ^ i67r“Z^ c** 2(2'7rWfl)‘’'^^ 2T^e^m,.Z^ g
^ AH 3^3 {kVy
THE HERTZSPRUNG-RUSSELL DIAGRAM 265
Substituting for according to (67) and remembering the defi-
nition of G{T) (Eq. [52]), we can write, instead of (69),
00 ) =
I i6t^Z^ gG{T)x\P
AH 2^3 n^{kTY‘^ ti^
^ - (70)
Comparing (70) with (64), wc can express 00) in the form
DIP
kPGi, 00 ) = {u ^ nP = hv\P/kf) , (71)
where
2n^
^ jyiZ) ^
^ , G{T) -xf )/ry ‘
Wc have the following alternative form for
n(X) _ n(^) M X'fl
(72)
(73)
Hence, the absorption cocITicient due to electrons in all the electronic
shells and also due to the free-free transitions can be expressed as
(74)
where
+ jyf ’ + . .
• • +
{h ^ «.) ,
( 75 )
> + . .
■ ■ + '>»'
(«. > ^ th) ,
(76)
=
> + . .
■ ■ +
(th > Mj) ,
(77)
Finally, if we consider a mixture of elements, we have to consider the
functions for each element defined as above; we then form the
266 STUDY OF STELLAR STRUCTURE
net absorption coefficient by weighting each of the values of
with the appropriate abundance factors. Thus we have
where
(78)
D(u) = 2 xzD(^iu) . (7^)
The summations in (78) and (79) are to be effected over all the ele-
ments.
It IS clear that D(u), as defined above, is a discontinuous function;
Fig. 22
stant between any two edges. Figure 22 illustrates the variation of
K, With frequency for the Russell mixture at T = i 4 x 10’ detrrecs
«d when log lG(T)m . 3. [The unitof ahsorpti™ cocnkiet ,
Figure 22 IS 3.89 X io“Sp7^*-s.] '-''-lu in
We have so far considered only the monochromatic absorption
THE HERTZSPRUNG-RUSSELL DIAGRAM
267
coefficients. We have now to calculate the Rosseland mean coeffi-
cient of opacity, as defined in equation (159) of chapter v. Intro-
ducing the variable u (defined as in Eq. [62]) in equation (159) of
chapter v, we find that
By (78) we have
I
K
(g“ - ly
°° e^u^du
(e" — i)^
I
K
1 c^'^u^du
D{ti) (e'* — 1)3
e“u^du
- i)^
(80)
(81)
By what has already been stated, the quantity D{u) is constant
between two absorption edges. Let the absorption edges (of all the
elements present), arranged in descending order, be u^, . . . . ,
Ui Then
/;(«) = D{Hi, «i , ,) (j/; > U ^ Ui^ .) . (82)
We can therefore express k in the form
i = N'' ~
K ^ IKUi, «f I .) ’
where 5 («) is the function defined by
S{u) =
X
"iOdil
(<!" — i)*
X
hUIu
(fi“ - 1)^
(8.0
(84)
'I'he integral in the denominator is easily evaluated. liy a partial
integration we have
c"uhin
(;;irir7y:
I
(85)
268
STUDY OF STELLAR STRUCTURE
The integral on the right-hand side has already been evaluated (Eqs.
[104] and [105], v) ; it has the value 71^/15. Hence, we have
Jo (e“ - 1)3 •
The function 5 (w) was first introduced by Stromgren, who also tabu-
lated the function sufficiently accurately for purposes of evaluating
the stellar opacity coefficient.
Stromgren expresses the opacity in the form
where Z)//, defined as in equation (64), is evaluated for A = 6,
and / is a numerical factor (the guillotine factor) depending on
[G{T)/Ne] and T. Numerically, it is found that
So far, we have neglected the ^-factor. Now the g-factors, strictly
speaking, enter as multipliers of the It is, however, clear
that only g-values near the absorption edges are of importance, for
g becomes different from (and smaller than) unity only far from the
absorption edge; in these regions, however, there will be an absorp-
tion edge of another element which will contribute to the stellar
opacity for the region considered. In other words, if we go to fre-
quencies V > then the particular (which contributes to
the absorption near v becomes small, compared to D{u).
An exception occurs for the absorption edges at high frequencies,
but for these the weight AS{u) soon becomes negligible. Stromgren
estimates that g = 0.90 will not lead to more than a 2 or 3 per cent
error in the final formula for the opacity. Hence, by (87) and (88)
we have
_ iSoNe
^ T^'H ■
(89)
THE HERTZSPRUNG-RUSSELL DIAGRAM
269
Table 12, due to Stromgren, gives values of Logio t for different
values of T and G{T)/Ne^ the computation having been carried
through for the Russell mixture. Thus, for any given electron con-
centration the table enables us to calculate the opacity arising from
a Russell mixture of elements.
TABLE 12
Guillotine Factor Logw t
T
log ic(r)/iv„i
03
6
S
4
2
iXiof '
0.03
2
.06
3
.02
4
.00
o.oO
0.13
<5
.01
.07
. 14
6
.08
• 14
. 22
0.3s
0.56
0.79
8
■25
•30
•37
•.SO
.68
0.93
10
•33
• 3 «
.45
.58
.76
0.99
j 2
•37
.41
.48
.60
•78
1 .01
14
•30
•43
•40
.61
• 70
1.02
16
.40
•43
• 40
.()0
.78
1.02
18
.40
•44
•40
• .s«
.78
1 .02
20
.41
■ 44
• 40
.60
0.70
1 03
215
0.48
0.51
0 . 5O
0 . ()()
We must now consider the effect of an admixture of light elements
(hydrogen and helium) with the Russell mixture. First of all, it is
clear that hydrogen and helium cannot directly contribute to the
stellar absorption coellkient, the essential reason being that the ab-
sorption edges of these elements lie in a spectral region the absorp-
tion in which region does not contribute (for all practical purposes)
to the Rosseland mean. The admixture of lighter elements has, how-
ever, an indirect effect.
Let us consider a mixture of elements hydrogen, helium, and
the Russell mixture with the abundance factors A", F, and
(i — A' — F ), respectively. Since, as we have seen, the lighter ele-
ments do not contribute to the opacity, the coefficient of opacity
is, accordingly,
K
(I ~
A
F) .
(90)
270 STUDY OF STELLAR STRUCTURE
On the other hand, we have (cf. Eq. [50])
N. = \fjii+X). (or)
Combining (90) and (91), we can write
^ ~l J3 s "I" ^ ^0 > (o-)
or, numerically,
« = 3-9 X io»s ■^)(i -X-V). ' (o.O
If the helium content is negligible, then
/c = 3 - 9 X
From (93) (or [94]) and Table 12 wc can calculate the stellar opacity
as a function of the density, temperature, and chemical conii)osi( ion
(here “chemical composition” is essentially equivalent to the abun-
dance of hydrogen and helium). Of course, Strdmgren’s table of the
“guillotine factor” i has been calculated for the case where the heavi-
er elements are assumed to occur in a delinite ratio. A cIosit e.\-
ammation shows, however, that this docs not materially alTcct the
formula for the stellar opacity.
We have so far restricted ourselves to photoelectric ionization as
contributing to the main source of stellar opacity. At high IcniixTa -
tures however (higher than in the interiors of the more coninnn,
stars), there is another physical process which becomes of impor-
tance in contributing to stellar opacity. The process in question is
the scattering by free electrons.
According to the classical electromagnetic theory, an accelerat.-.l
e ectron emits radiation; we are here concerned with the converse
phenomenon. J. J. Thomson has given for the scattering coetlicient
per electron the formula
(b.s)
THE HERTZSPRUNG-RUSSELL DIAGRAM
271
It will be seen that <r is independent of the frequency. Using (91) for
the expression of the number of free electrons, we have for the con-
tribution to the mass absorption coefficient by electron scattering
the expression
Kv(v)dv
2 11
(96)
or, numerically,
KeW = o. 2 op(i + X) .
(97)
If electron scattering were the only source of stellar opacity, then
the Rosseland mean coefficient Kc would be given by (cf. Eqs. [80]
and [85])
L - 11 r°° I
Kn 4^Vo icc{u) — i)‘^
(98)
Since, however, Kc{n) is independent of 7/, the integral in (98) is
easily evaluated. It is found that
O’" — iy~~ 2 ^ I + + 3“’^ + • • • J ’
'rhe numerical value of the right side of the foregoing equation is
found to be 1.055. Hence, by (97), (98), and (99),
o. 20p(l + A')
1-055
= o. i9p(i + X) .
(100)*
If photoelectric ionization and electron .scattering are both about
equally im[)orlant, then we must form the Rosseland mean of the
combined absorption coefficient, which by (78) and (97) is
«.H i(») = + *:-’(«) • (loi)
Thus, the resulting coefficient of opacity is given by
T _ ji_5 r°^ T u^e^'Uiu . .
Kr\i D{tl)u~'^ + Kr{u) (C" — ly'
It is lurcssiiry to distinguish In-twccn and Kg.
272
STUDY OF STELLAR STRUCTURE
Stromgren has evaluated /Ce+i according to (102) for a number of
typical temperatures and densities at which electron scattering be-
comes important, and he has given the following empirical rule :
For a given T and [G{T)/N^, calculate the coefficient of opacity
Ki due to the ionization process only, using the table of guillotine
factors. Then calculate /Cc, which is given by (100). The actual opac-
ity /Ce+i is equal to the greater of the two quantities k* and Ke plus 1.5
times the smaller of the two.
This completes the discussion of stellar opacity.
S- Determination of the mean molecular weights of the stars . — We
have shown in the last two sections how the mean molecular weight
and the stellar opacity can be determined in terms of the abundances
of hydrogen and helium. In the discussion we shall in the first in-
stance assume that the helium content can be neglected. We shall
return to this question in § 9.
Our assumption, therefore, is that the stellar material is a mixture
of hydrogen and the heavier elements in the ratio, by weight,
Xo : I — Xo- We shall further assume that the heavier elements
form a Russell mixture. Actually, we make this definite assumption
about the heavier elements because that is the ratio in which
the elements are approximately present in the sun and in stellar
atmospheres (Russell and C. H. Payne), and which further
happens to agree roughly with the abundances with which these
elements are present in the earth^s crust. But this assumption,
made for the sake of definiteness, is from the point of view of
stellar interiors a very ‘‘harmless one,’’ in the sense that any
other assumption regarding the abundances of the heavier elements
will lead to substantially the same conclusions regarding the abun-
dance of hydrogen. This is seen when we compare our first and sec-
ond approximations for /x, which are given in § 3. On the first ap-
proximation we derived, with no particular assumption regarding
the abundances of the heavier elements but only assuming that the
material is highly ionized, that
2
I + 3X0 ’
(103)
THE HERTZSPRUNG-RUSSELL DIAGRAM
273
while a more rigorous consideration of the state of ionization for
the case when the heavier elements form a Russell mixture, led to
** ^ + nitii - A„) ’
where hr is tabulated in Table 10. An examination of (104) with
the values of n/e given in Table 10, and a comparison with (103),
readily shows that the actual specilication of the abundances of the
heavier elements is hardly of importance in the present connection.
The same thing is true of stellar opacity, for which we can use
K = 3 ■ 90 X 10=5 i (i _ x,=) . (los)
It should be mentioned in this connection that if, instead of the
Russell mixture, we use one in which there is a higher proportion of
the heavier elements, then, though the coefllcient in the opac-
ity formula (Kqs. [64], [71], [73], and [74]) increases, this elTect is
largely compensated by a corresix)n(ling increase^ in the mean mo-
lecular weight, which works in the opposite direction. We thus see
that equation (105) is valid over a wide range of the relative abun-
dances of the heavier elements.
We shall now proceed to outline the method of determining the
hydrogen abundance, A'„, for a star of known mass, radius, and lu-
minosity.
We shall write equation (105) in the form
« = «» 7 7-;.5 . (106)
where
«.. = 3-9 X io' 5 (i — ,V;i) . (107)
® ''riiiil llicTC* is an increase of ju is seen as follows: lM)r a Russell mixture we have sc‘cn
that when it is completely ionized there are 0.54 free particles per unit atomic weight.
If there is a higher proportion of the heavier elements than in the Russell mixture, there
will be a smaller number of free particles per unit atomic weight; this will increase g,
and therefore decrease the number of free electrons per unit volume.
274
STUDY OF STELLAR STRUCTURE
The formula for k which was used in § i (Eq. [3]) is of the same form
as (106), above. We can therefore use the luminosity formula (14),
which can be written as
ifS-S
K * = 7- 17 X 10="' , (108)
where L, M, and R are expressed in solar units. Further
/£* = 7 , (109)
where Hs a mean value for the guillotine factor (cf. Eq. [8]).
Thus, for a star of known Z, ilZ, and R and an assumed value
for iu, equation (io8) suffices to determine k* and if we can estimate
the value of then we have — so to say — an astronomical determina-
tion of the physical constant /Co. Again, an assumed value for /z im-
plies, according to (103) or (104), a definite value for Xol hence we
have, according to (107), a physical determination of /Co. We now
arrange by a proper choice of ji (or Xo) that the astronomical and
the physical values of Kq agree. The value of Xo (or fx) which brings
about this agreement determines the hydrogen content of the star
under consideration. The point which remains to be settled is the
determination of the mean value I of the guillotine factor.
According to the discussion in § 4, / depends on [G{T)/N^],
which, according to equation (36), is defined by
G{T) _ (27rWo^T)^/* I
A. “ ^ hi Ne'
(no)
To determine I it is clearly sufficient to consider the first approxima-
tion of § 4, according to which
I H” 3“^° ^ nn
= i(i + ^») ^ .
By (no) and (in) we have
G{T) _ I + 3X0 kT
Ne ^ hi i+Xo
THE HERTZSPRUNG-RUSSELL DIAGRAM
275
■which is easily seen to be equivalent to
G{T) ^ 3 I + 3-^0 I - ^
Ne h? a I -|- A(i (8
Numerically, the foregoing is
or
G{T)
Ne
2 . 63 X 10"* X
I + 3-^0 I — /3
l+Xe p
T-3/\
(II4)
log
r , T 1 + 3^0
2.303I 14.42 + Log
(IIS)
Since, according to our first approximation considered in § 7 of
chapter vi, the standard-model density distribution is to be regarded
as a first approximation, we can use for jS in the foregoing equation
the value determined by Eddington’s quartic equation. We shall
then have to study the march of [G( 7 ’)/A^,.l through the star and
by using the table of guillotine factors (Table 12) infer the appro-
priate mean value, 1. We shall illustrate the estimation of I for the
sun. If we assume for ii the value T.05, the solution of the quartic
equation yields 1 — (3 = 0.004, by equation (62) of chapter vi,
it is found that T,. = 20,000,000 degrees. Table 13 (due to Strom-
TAHLK 1 .^
T
K,rf^i
t
T
t
20X io‘*
2.8
7
12X\0^
3-5
3 .«
A I
5
A
iS
2 . y
6
10
i(>
3 • ^
()
8
4
3
2
14
3-3
5
()
*+ ■ *
4.^)
gren) gives the corresponding variation of / through the star. From
the table Stnimgren estimates that the ai)propriate mean value of /
is about 5, which is seen to be the value of / at about two-thirds of
the central temi)erature. I’his appears to be a general rule, so that
276
STUDY OF STELLAR STRUCTURE
to determine the order of magnitude of I we first determine ac-
cording to equation (62) of chapter vi and then calculate {G{T)/N^
for T = 2/iTc as given by (114) or (115). From Table 12 we then
obtain the corresponding value of t. This value of t is used in the
luminosity formula as a first approximation to 1 . After a first ap-
proximation to ju has been obtained, the quantity I having been esti-
mated in the foregoing manner, we can then proceed to a second ap-
proXjimation by determining I defined appropriately (cf. Eq. [8]).
Using this method, Strdmgren has computed the values of ii and
has thus inferred the hydrogen contents of those stars for which
there is fairly reliable information concerning Z, and R, Tables
i^a and 14& illustrate the determination of fi for Capella and the sun.
TABLE 14 a
The Determination of the Hydrogen Content of Capella A
Xo
A*
I -/9
Lok *:o (Astro.)
Log Ka
(Physical)
0.34
0.9s
0.04
25-37
25.53
.31
I .00
.04
25-51
25.54
.28
I. OS
.05
25.64
25.55
.25
1 . 10
.06
25.76
25.56
0.22
I 15
0.07
25.88
25.56
log [G(T)/Ay = 7; 7 =i; X'„=o.3o; iu=i.oi
TABLE 14b
The Determination of the Hydrogen Content of the Sun
Xo
I-P
Lok Ka (Astro.)
Lor kq
(Physical)
0.36
I .00
0.003
24.85
24.84
.33
1.05
.004
25.01
24. «5
0. 2 Q
I . 10.
0.004
25. 16
24.86
log [G(r)/iVJ = 3; J«=5; Ar„=«o.36; m=i.oo
In connection with the foregoing solutions it is necessary to re-
mark that the values of Xo given do not represent the only solution
to the problem. For a given star there are, in general, two values of
THE HERTZSPRUNG-RUSSELL DIAGRAM
277
Xa for which there is agreement between the astronomical and the
physical values of k*. The second solution, as we shall see presently,
however, corresponds to an extremely high abundance of hydrogen.
We can best illustrate the existence of this second solution in the
case of Capella, for which the guillotine factor is approximately
unity.
If there exists a solution corresponding to which X„ is almost
unity, we can put m = 0.5 in the luminosity formula. Again, for the
order of stellar masses we shall normally be interested in (masses
less than 10 O), / 3 c is very nearly unity. Hence, we can also put
18c = I. Equation (108) now reduces to
K* = 4.0 X 10” (jx = 0.5, = i) • (108O
For Capella, M = 4.18, L = 120, and R — 15.8. Inserting these
values in (108'), we find that k* = 2.17 X It is also found
that I is unity, and hence
while
K„ (astronomical) = 2.17 X 10*’ ,
Ka (physical) = 3.89 X lo-'-’ (i — Aj|) .
From the foregoing it follows that .V„ = 0.997. In other words, this
second solution corresponds to 99.7 per cent abundance of hydrogen,
while our first solution corrcsjxmds to 30 per cent of hydrogen.
Similar results will be obtained for the other stars. Hence, quite
generally, the second solution corresjwnds to an extremely high
abundance of hydrogen, and it is improbable that such an extreme
abundance of hydrogen can correspond to reality. Actually, there
are reasons to believe that the hydrogen abundance in stellar in-
teriors must be less than in stellar atmospheres' - the e.ssential ground
for this belief being that the heavier elements will “sink” relatively
more toward the center of a star than the lighter elements, and it
appears that hydrogen is not present in stellar atmospheres to any-
thing approaching 99.7 per cent by weight. We shall therefore re-
strict ourselves (unless otherwise slate<l) to the solution which corre-
sponds to a “moderate” abundance of hydrogen.
278
STUDY OF STELLAR STRUCTURE
6. General remarks. — Before we proceed to describe Stromgren^s
results for other stars and the bearing of these calculations toward
an interpretation of the Hertzsprung-Russell diagram, it is neces-
sary to make some comments concerning our present attitude, as
compared with that generally adopted in an earlier epoch (i.e., prior
to Strdmgren’s systematic work).
In Eddington’s earlier work it was assumed that all stars have
the same mean molecular weight (m 2). This implies, for all prac-
tical purposes, that the lighter elements (hydrogen in the present
connection) are not abundant. The assumption of constant fx is not
only characteristic of Eddington’s early work but has been implicitly
assumed quite generally. This assumption, according to our present
point of view, has to be abandoned. The reasons can be briefly sum-
marized as follows:
Let us suppose that the abundance of hydrogen is negligible. Then
we immediately come into conflict with the physical theory of the
stellar opacity. The nature of the conflict can be illustrated by tak-
ing the case of Capella. Observationally, we have
L = i2oLq ; M = 4. 18 O ; R — 15. 8 /?q .
Let us assume (as in Eddington’s early work) that m = 2.1 1 . Then
we have
jS = 0.717 ; Tc = 7.9 X lo'"’ .
The guillotine factor I is found to be unity, so that, according to
(107), (X„ = o),
/c* (physical) = 3 . 9 X .
From (108), on the other hand, it is found that
K* (astronomical) = 8.8 X 10"^. (116)
We see that the two values of k* differ by a factor of about 23; this
is the famous ^'opacity discrepancy.” In spite of this discrepancy,
the tendency was to assume that the origin of it is due to the in-
adequacy of the physical theory, and in Eddington’s early work /c*
was assumed to be equal to its ‘‘astronomical value.” The luminosity
THE HERTZSPRUNG-RUSSELL DIAGRAM
279
formula (108) (with k* according to [116]) was therefore used to
predict the luminosities of other stars. It was found that the lu-
minosities thus predicted agreed with observation. This was taken
to imply that the stars in fact form a one-parametric sequence of
configurations.
But the foregoing point of view has to be abandoned, for the
more refined theory of the stellar opacity now available leaves no
room for doubting the physical theory. Consequently, we must ac-
cept an abundance of the lighter elements, in particular of hydrogen,
to remove the discordance between the physical and the astronomi-
cal value of K*. But now it may be argued that we may allow for an
abundance of hydrogen but still use a constant ju for all the stars, so
that the luminosity formula can still be used to predict the luminosi-
ties for stars of known mass and radius. This idea gained some cur-
rency when it was found that both Capellaand the sun lead to about
the same value of A"«. But one important dilTerence has to be noticed.
We obtain the same value of -Yo for Capella and the sun because
of the guillotine factor. For the sun the guillotine factor is 5, while
for Capella it is unity, so that, if we use for k* the value derived
from Capella, and use (ro8) to predict the luminosity of the sun, we
should be wrong by a factor of 5, *fJow, this same argument (due
originally to Eddington) can be employed to show the necessity for
introducing a variable .V„. Consider, for instance, the sun and .f Her-
culis A.-^ Ik)lh have very nearly the same mass (Mf urr = 0.96), but
Herculis A has a radius about twice that of the sun and a lu-
minosity about four times that of the sun. Suppose we assume that
the sun and f Herculis A have Ihe same value for /i. It is found now
that for Herculis I is 2.,^. Again, since J* Herculis has twice the
solar radius, the predicted luminosity would be
A (j)re(licled)
2.3 X ^ oc-
2»-s X s ■ ’
while L (observed) =4.0 i.e., a discrepancy of a factor of 16. Thus,
f Herculis must have a dilTerent value of g from that of the sun.
Indeed, calculation shows that for f Herculis, A"„ = o.ti and
M = i'45-
i 'riiis is (inly un cwiimplc*. Wc can give other similar examples.
28 o
STUDY OF STELLAR STRUCTURE
Thus, for consistency we are forced to accept a variable ix for
stars; consequently, the luminosity formula (io8) has to be used
to determine /a (or for individual stars, rather than to predict
the luminosities for different stars. It is necessary to emphasize
this because Eddington, who, independently of Stromgren, intro-
duced the abundance of hydrogen to remove the ‘^opacity discrep-
ancy” for the case of Capella and the sun, seems inclined to the view
that the luminosity formula can still be used to predict L for other
stars, using a constant /a ^ i.o for all stars. The objection to using
the luminosity formula in determining for the individual stars
seems to arise from an uneasiness that ju is used as an ''adjustable
parameter” to "save” an "inadequate” theory. But the answer to
such an objection is that, if we allow fx to be variable, the derived
/x’s show a systematic variation in the plane of the Hertzsprung-
Russell diagram and do not show any randomness. In other words,
we derive an interpretation of the characteristic features of the
Hertzsprung-Russell diagram in conformity with the Vogt-Russell
theorem.
7. Interpretation of the Hertzsprung-Russell diagram , — According
to the method outlined in § 5, it is relatively simple to determine
the hydrogen contents of stars for which the values of the funda-
mental parameters Z, ilf , and R are known. The computations have
been carried out for about forty stars, and the resulting hydrogen
contents for some of them are given in Table 34 in the form of an ap-
pendix. We shall here be concerned only with the general results,
but it may be mentioned that for the B stars it is necessary to take
into account the effect of electron scattering, which was considered
at the end of § 4.
First of all, the question arises whether we cannot represent the
whole observational material — ^within the limits of the uncertainty
of the observations — on the assumption of a constant /x for all the
stars. We have already considered this question in § 6, and a closer
examination now reveals that, though the most commonly discussed
stars — the sun, Capella A, and Sirius A — have about equal hydro-
gen content (^35 per cent), it is yet not possible to predict the lu-
minosities of all the stars considered on the basis of a constant ix.
If an attempt is made, we encounter discrepancies in the predicted
THE HERTZSPRUNG-RUSSELL DIAGRAM
281
luminosities sometimes amounting to as much as by a factor of 50,
and further, these discrepancies show a systematic character. The
magnitude of these discrepancies and their systematic nature pre-
clude the possibility of constant n- It is highly satisfactory that
we are led to a variable n (or Xo) purely from observations, for we
are led to precisely the same conclusion by appealing to the Vogt-
Russell theorem. For, two stars of equal mass can differ (in radius
Kio. 23^7.— Kach (lot represents a star and is labeled by the comiiutcd .Vo (Strdm-
grcn,Z.v./. /!/>., 7, 222, 1033).
or/and luminosity) only on account of a dilTcrencc in chemical com-
position, and wc have now to examine whether the observed masses,
radii, and luminosities, and the derived hydrogen contents enable
us to arrange the stars as a two-parametric family of configurations
(the two parameters being M and A'„).
In Figures 23a and 236 wc have plotted Log M against Log R.
Each star is represented by a point in this plane, and we label each
point by the appropriate .Y„. We see that the plot in this diagram
enables us — more or less unambiguously -to draw curves of constant
282
STUDY OF STELLAR STRUCTURE
fj, (or Xo) in the (Log M, Log R) plane. Schematically, the situation
that arises is shown in Figure 24.
It is clear from Figure 23 or Figure 24 that, if we consider a se-
quence of stars of a given M but of increasing radii, then along this
sequence the hydrogen content decreases, while the mean molecular
weight increases. This is, indeed, a quite general result: For a given
Fig. 236. — This is a revised diagram in which the more recent data have been used.
As in Fig. 23a, each dot represents a star and is labeled by the computed A'».
mass^ with increasing radius^ the hydrogen content decreases. This re-
sult is easily understood. Observationally, it is well known that a
rough empirical mass-luminosity correlation exists in nature.- An in-
crease of radius, then, has two effects: first, the guillotine factor
decreases, and, second, the radius factor in the luminosity formula
increases. Both these effects act in the same sense-- toward lowering
the predicted luminosity; to counteract this effect it is necessary to
increase or, what comes to the same thing, to decrease the hydro-
THE HERTZSPRUNG-RUSSELL DIAGRAM
283
gen content. In the case of the massive stars, however, another effect
becomes important: For the massive stars of smaller radii (i.e., for
the B stars which form the continuation of the main series) the cen-
tral temperature is sufficiently high to reduce the magnitude of the
general (i.e., the Kramers- Gaunt) opacity, thus making the con-
tribution to the absorption by electron scattering important. Thus,
while a decreasing radius still corresp>onds to an increase in the guil-
Ki<;. 24. TUv. scmifiupirlnil riirvrs of rnnslanl .V„ in llu* (I.of' 4 /, Log R) plane
(Slnimgren, Xs. f. A/)., 7, 222, ig.s.O- Laeli nirve is lahelnl l»y the corresi)()n(ling
value of .V„.
lotine factor, the increasing imi)orlance of electron scattering with
decreasing radius acts in the opposite (lirrclion, in counteracting the
decrease of stellar opacity arising from the guillotine factor. Hence,
in the case of the massive stars, though decreasing radius still corre-
sponds to increasing .V„, the range of variation in .V„ for given
change in R is much less tlian for stars of “ordinary" masses (i.e.,
M < 4 O).
Once we have drawn an empirical set of curves of constant .V„ as
in Figure 24 (which, it will be remembered, combines the results de-
rived from a theoretical \L, M, R, g] relation and the set of values
of L, M, and R that occur in nature), then, if we si)ecify the mass
284
STUDY OF STELLAR STRUCTURE
and the radius of a star, the appropriate /t or Xo can be directly read
off from the diagram; from a knowledge of M, R, and /t we can pre-
dict L. Further, we can transform the curves of constant Xo from
the (Log M, Log R) plane to a set of curves of constant Xo in the plane
of the Hertzsprung-Russell (for short, “H.R.”) diagram. As is well
known, the co-ordinates which describe a star in the H.R. plane are
the absolute magnitude (essentially —2.5 Log L) and the spectral
type (essentially Log T^), Te (the effective temperature) increasing
Fig. 25. — ^The curves of constant X, (the full-line curves) and the curves of constant
M (the dotted curves) in the plane of the HertssprunR-Russell diagram (Strdmgren,
Zs . f . Ap ., 7, 222, 1933).
toward the left. To transform the curves of constant A'o from the
(Log M, Log R) plane to curves of constant A'o in the H.R. plane, we
go along each particular curve in the (Log M, Log R) plane, and for
each point we calculate L according to the luminosity formula, and
hence also Log T^; for, according to the definition of the effective
temperature, we have
Similarly, we can draw curves of constant M in the H.R. plane.
These two sets of curves will enable us to determine both the mass
and the hydrogen content of a star merely from its position in the
THE HERTZSPRUNG-RUSSELL DIAGRAM
285
Hertzsprung-Russell diagram (see Fig. 25). It may happen that the
curves of constant Xo intersect in the H.R. plane. In such cases ref-
erence must be made to the (Log M, Log R) plane. We have thus suc-
ceeded in arranging the stars in a two-parametric sequence entirely
in conformity with the requirements of the Vogt-Russell theorem.
From the topography of the curves of constant X„ and constant
M, we derive the following interpretation (due to Stromgren) of the
characteristic features of the Hertzsprung-Russell diagram:
The main series up to spectral class A is the locus of stars of hy-
drogen content varying between 25 and 45 per cent — ^i.e., about a
mean of 35 per cent — and masses running up to 2.5 O. Stars of
small mass and low hydrogen content are relatively rare — they oc-
cur as subgiants of spectral classes G-K. The gap between the M
giants and the corresponding dwarfs (on the main series) arises from
the circumstance that not even stars of low hydrogen content “scat-
ter” in this region. The massive stars (Af > 5 O) occurring in the
region of the B stars which arc rich in hydrogen (A'„ sometimes going
up to 95 per cent) form the continuation of the main series— the
continuation arising from the circumstance that massive stars with
“medium" hydrogen content (0.4 < .V„ < 0.8) which arc on the
main series occur in a very small region of the H.R. diagram between
the B and the A stars. (We shall obtain evidence in chapter viii for
the breakdown of the standard model for the very massive stars.
Further, along the main series the breakdown probably sets in at
about M = 10 O. 'File investigations of the hydrogen content of
the B stars is therefore somewhat inconclusive. We shall return to
these matters in chapter viii.) 'I'he giant branch is characterized by
stars having about the same hydrogen content as (or somewhat less
than) the main series stars. 'I'he giant branch is limited on the side
of low luminosity, since stars of low luminosity are relatively rare.
On the side of high luminosity it is limited again, because, for X„ a
little greater than 0.3, the characteristic bend of the curves of con-
stant .V„ disappears, and also because the stars of large mass with
hydrogen content greater than about 40 per cent scatter over a large
area in the H.R. diagram, which must, therefore, be sparsely popu-
lated. 'I'he gap (the “Hertzsprung gap”) in the giant branch in the
region of spectral class F is probably due to a real scarcity of stars
286
STUDY OF STELLAR STRUCTURE
with masses between 2.5 and 4.5 G. The supergiants, then, are in-
terpreted as massive stars with medium hydrogen content. The
‘^spreading-out’’ of the curves of constant Xo in the supergiant region
of the H.R. diagram is easily understood from the remarks made
on pages 282 and 283.
5.0 4.8 4.6 4.4 4.2 4 0 3.8 3.6 3.4
Fk;. 26 — The iibscissac are Log 7 V,* the ordinutes are al)solute holonirlrie magni-
tudes (Kiiiper, Ap. 7 ., 86, 176, 1937). The clusters are idenlilled in Table 2 of Kuiper’s
paper.
8. Ktdper^s interpretation of the cluster diajirams: the hydroiicn con-
tent of the Ilyades stars . — So far wc have considered only the charac-
teristic features of the general H.R. diagram. It is clear that we can
construct the absolute-magnitude--spcctral-typc diagrams, includ-
ing in the plot only such stars as are physically associated like the
stars in a cluster. The H.R. diagram for stars in a cluster may be
called a “duster diagram.” The pioneering work on this subject was
THE HERTZSPRUNG-RUSSELL DIAGRAM
287
done by Hertzsprung in 1911, and extensive studies of a systematic
nature of the cluster diagrams have been carried out by Trumpler;
for a general discussion of the subject from the observational side,
and for references, see a paper by Kuiper quoted in the Bibliographi-
cal Notes at the end of this chapter. Figure 26 is taken from Kuiper^s
paper. From the similarity of these curves with the Stromgren
curves of constant A",,, Kuiper infers that the stars in a cluster are
characterized by approximately the same hydrogen content.
A comparison of the diagrams for the Pleiades and the Hyades in-
dicate that Hyades stars should have relatively low hydrogen con-
tent. In the case of the Hyades cluster, Kuiper’s suggestion is capa-
TABLIi: 15*
'I'lIK IIVDROCKN CONTICNT OK TIIK STARS IN IIVADKS
ADS
boK L/Lq
1.0K K/Rq
I,.IK (J//G)
A'o
WeiKht
5264
-fo . 6<)
+ 0. 1 1
-1-0.07
<0.23
li
4- .40
4- -OO
+ .04
•30
r
5135
+ .07
“ ,o?)
- . O).?
3.
~\- . 0()
- .0.^
— .21
< .13
1
2
■WS
— .of»
— . O)
• 15
3
32JO
-0.37
-0 44
0.02
2
* For ADS Kuiper has revised the data on llie basis of ueHilional information. The new
value has been used here.
ble of verification. There are six stars in this cluster for which Kuiper
has derived the values of A, A/, and R. Iu)r these stars Strtimgren
has computed the hydrogen content and the results are given
in 'fable 15. 'fhe uncertainties in the values of A, A 4 , and R used
arise essentially from the uncertainties in the j)arallax of the individ-
ual stars (though the i)arallax of the cluster itself is fairly reliably
known). According to Kuiper, the weighted mean of the six de-
terminations, which gives A',, (mean) = 0.16, may be considered as
a reliable estimate of hydrogen content of the Hyades stars.
9. The abundimee of helium in stellar interiors. We found in § 3
that for purposes of the analysis of stellar interiors it is siilTicient
to consider the abundances of hydrogen, helium, and all the other
heavier elements (Russell mixture) lumped into one group. In the
discussions in §§ 5, 6, 7, and 8 we assumed that the helium content
288
STUDY OF STELLAR STRUCTURE
could be neglected. This assumption is justifiable in the first in-
stance in so far as investigations of the stellar atmospheres seem to
indicate that hydrogen is very much more abundant than helium.
But it should be remembered that the determination of the abun-
dances of hydrogen and helium in stellar atmospheres is a matter of
great complexity. This has to be borne in mind, especially in view
of the fact that in a recent investigation on the transmutation of
elements in stellar interiors (chap, xii) by von Weizsacker the sug-
gestion is made that helium may be very much more abundant than
all the other heavier elements put together. Indeed, von Weizsack-
er’s theory seems to require helium to be as much as eight to ten
times as abundant as the Russell mixture; and Stromgren has ex-
amined whether this requirement of von Weizsacker’s theory is com-
patible with the data concerning the masses, luminosities, and radii
of the stars. We shall follow Stromgren’s discussion of this matter.
Let us consider the ^‘second solution’’ for the hydrogen content,
the existence of which we pointed out in § 5. In the case of Capella
we found that the ‘^second solution” corresponds to an abundance of
99.7 per cent of hydrogen. But this extreme abundance of hydrogen
corresponds to the relative abundance of hydrogen to the Russell
mixture which is so high as to be quite improbable. However, if
we reduce the amount of hydrogen by a small amount and replace
it by helium, then we increase ix so that the predicted luminosity, in
the first instance, is greater. Hence, in order to predict the correct
luminosity, we must increase the absorption and hence increase the
abundance of the heavier elements. It is clear that by a suitable
increase of the amount of the Russell mixture present we can again
obtain agreement between the observed and the predicted luminosi-
ties. We now have two unknowns — the hydrogen content, A", and
the helium content, F. There is only one relation — the theoretical
(L, M, R, fj) relation — available, so that we can determine the hydro-
gen content X and the ratio of Russell mixture to helium {U : Y)
as functions of the helium content. Tables i6a and 16b (due to
Stromgren) illustrate the results of such calculations for the case of
the sun and Capella.
The table shows that the (L, M, i?, ju) relation can be used to de-
termine the maximum value of the ratio Y : U. Also, a definite
THE HERTZSPRUNG-RUSSELL DIAGRAM
289
physical theory of the transmutation of elements in stellar interiors
would provide a theoretical value for the ratio V : U. For a specified
value of the ratio Y \ have, in general, two solutions, but it
will be more difficult to decide between them now. For example, if
the ratio were prescribed to be 10, then for the sun we have for the
relative abundances of hydrogen, helium, and the Russell mixture
TABLE I6a
The Hydrogen and Helium Content of the Sun
Helium
Content (F)
HydroKcn
Content (X)
Russell-Mixture
Content (U)
y : U
U : X
0.00
1 .00
0.002
0
0.002
• og
0.90
.004
22
.005
.19
0.80
.OOQ
21
.011
.28
0.70
.02
IS
.026
•36
0.61
.04
10
■059
.42
o-Si
.07
6
.14
0.43
0.43
0. J4
3
0.32
TABLE 166
The Hydrogen and Helium Content of Capklla A
Helium
Content (F)
Hydrogen
Content (.Y)
Russell-Mixture
Content {£/)
F : 1 /
i/ : X
0.00
1 .00
0.00 1
0
0.001
.10
0.()0
.002
4 ()
.002
• 19
0.80
.005
37
.006
■ 2g
0
0
.01 2
24
.02
•37
0.()O
•025
>S
.04
•42
051
.07
(>
.13
0.47
0.42
0.11
4
0 . 26
the two solutions 60 : 36 : 4 and 90 : 10 : 0.3. Of the two so-
lutions, the first is probably more consistent with the spectroscopic
evidence from the study of stellar atmospheres. It should further
be noticed that the observational uncertainties in L affect the de-
rived content of the Russell mixture directly, so that the chemical
composition derived on the hydrogen-helium Russell-mixturc hy-
pothesis is very much more sensitive to the uncertainties in the ob-
servational values of L than is the case on the hydrogen- Russell-
mixture hypothesis.
290
STUDY OF STELLAR STRUCTURE
For the other stars we have similar results; for Capella A, Strom-
gren finds that the calculations lead to results very similar to those
for the sun. For the subgiants, it again appears that they are rela-
tively poorer in hydrogen than the main-series stars. Further, the
connection between the radius and the content of the heavy ele-
ments appears to be the same as in our earlier discussion on the basis
of the hydrogen-Russell-mixture h3rpothesis. Indeed, the run of the
curves of constant abundance of the heavy elements in the plane
of the H.R. diagram derived in § 7 (Fig. 25) is seen to be very gen-
eral: If we consider stars of ^‘medium” masses {M < 2.5 O), then
an increase in R has two effects : it decreases the guillotine factor /,
and it increases the factor in the luminosity formula. Both these
effects, acting in the same sense, make the predicted luminosity too
low in the first instance (there being an empirical rough mass-lumi-
nosity correlation) . To obtain agreement between the predicted and
the observed values of Z, we must increase the abundance of the
heavier elements; i.e., U increases with increasing R, For the more
massive stars, on the other hand, the electron scattering — now of im-
portance — acts in the opposite direction to the guillotine factor and
the factor in the luminosity formula. The ^‘spreading-out” of
the curves of constant U in the H.R. diagram is thus seen to be very
general. The general conclusions, then, are essentially the same as
before. The important point to note in the present connection is
that an abundance of helium comparable to that of hydrogen is
“compatible with the observational data concerning the masses, lu-
minosities and radii of the stars” (Stromgren).
BIBLIOGRAPHICAL NOTES
This chapter deals almost entirely with the results contained in —
1. B. Stromgren, Zs.f. Ap, 4, 118, 1932.
2. B. Stromgren, Zs.f. Ap., 7, 222, 1933.
3. B. Stromgren, Erg. exakt. Naturwiss.^ 16, 465, 1937 (§§ 16, 17, and 18 of
this paper).
§ I. — For the whole discussion the luminosity formula L = — jd,)
is made the fundamental starting-point. The exposition of the thcor\’
starting with the luminosity formula makes the presentation rather neat, and
this particular arrangement of the arguments is believed to be new.
§ 2. — 4. H. Vogt, A.N.^ 226, 301, 1926.
THE HERTZSPRUNG-RUSSELL DIAGRAM
291
5. H. N. Russell, in Russell, Dugan, and Stewart, Astronomy ^ 2, 910,
Boston, 1927.
§§3 and 4. — References i and 2. Also —
6. R. H. Fowler and E. A. Guggenheim, M.N.^ 85, 939, 1925.
7. A. S. Eddington, M.N,, 92, 364, 1932.
§§ 5, 6, and 7. — References i and 2. Also —
8. A. S. Eddington, M.N.^ 92, 471, 1Q32. In this paper Eddington tends to
a belief that all stars have the same hydrogen content. 'Phis diilers from the
point of view currently adopted.
§8. — 9. E. Hertzsprung, Potsdam Pub., 63, 1911. This classical paper,
which is very rarely quoted, contains a summary of Hertzsprung’s earlier work
(1905-1909), in which the giants and dwarfs were discovered. It gives a clear
description of the main series (Hauptserie) as being a group of stars of nearly
the same radii but widely different surface temperatures, which results in the
observed differences in luminosity. Giants and supergiants (c stars) are also
discussed. The paper contains the first diagrams relating color equivalent or
spectral equivalent with absolute magnitude.
10. R. J, Trumpler, Pub, A.S.P., 40, 265, 1928.
11. G. P. Kuiper, Harvard Bidl., No. 903, 1936.
12. G. P. Kuiper, Ap. 86, 176, 1937.
§ 9. — B. Stromgren, Ap. 87, 520, 1938. Also reference 3
'["he following further references may be noted:
13. R. H. Fowler, Statistical Mcciianks, 2d ed., chaps, xiv, xv, xvi, and
xvii, Cambridge, 1936.
14. S. Rosseland, Aslrophysik anf Atomilicorciisrhcr Gnaidlai’r., Ilcrlin, 1931
(§ 16 of this book).
In references 13 and 14 problems connected with “excluded volumes,”
“electrical pressure,” etc., are considcrc<l -toiiics which have not been treated
in the monograph.
CHAPTER VIII
STELLAR ENVELOPES AND THE CENTRAL
CONDENSATION OF STARS
In this chapter we shall discuss the equilibrium of stellar en-
velopes. By a ^‘stellar envelope'* we shall mean the outer parts of a
star, which, though containing only a small fraction (for definite-
ness, we shall assume this fraction to be lo per cent) of the total
mass, M, nevertheless occupy a good fraction of the radius, R, A
study of stellar envelopes has a twofold importance for astrophysical
theories: first, it extends the region of the study of the conventional
stellar atmospheres into the far interior, and second, it has also a
very definite bearing on the studies of the deep interiors which are
our main concern in this monograph. Thus, the central condensa-
tion of a star, defined as the fraction of the radius, R, which in-
closes the inner 90 per cent of the mass, M, must give some indica-
tion of the concentration of the mass toward the center of the star
under consideration. It is clear that (i — f*) is a measure of the
extent of the stellar envelope. The main problem which we shall
consider in the theory of stellar envelopes is the evaluation of the
central condensations of stars of known L, Af, and R and assumed
chemical composition. We shall see that this subject is closely re-
lated to the problems discussed in chapter vii.
I- The equilibrium of stellar envelopes . — The general theory pre-
sented here is due to Chandrasekhar.
In writing down the equations of equilibrium, the following two
simplifications will be introduced: (a) that there are no sources of
energy in the stellar envelope, and (6) that the mass contained in
the envelope can be neglected in comparison with the mass of the
star as a whole. Indeed, these two assumptions can be taken to
define the stellar envelope.
The equations of equilibrium of the stellar envelope, then, are
d{Po + Pr) _ GM
dr r* ^
292
STELLAR ENVELOPES AND STARS
293
and
dpf kL
dr 47rcr“ ^ ’
(2)
the sjonbols having their usual meaning. The formula for the stellar
opacity appropriate to the present discussion is (cf. Eq. [94], vii)
K = 3.89 X 2^- (3)
It is found that, under the circumstances of the stellar envelope, the
guillotine factor t does not vary appreciably; in many cases of prac-
tical importance it is very near unity, and even under the most un-
favorable circumstances it varies only by as much as a factor of 3.
Further, the guillotine factor occurs in the final formulae (which de-
termine the central condensation ^*) only as a square root, so that
we can conveniently replace ^ by a constant le throughout the en-
velope. We shall therefore write (3) as
K = KopT s ;
Ko —
3.9 X - XI)
h
Dividing (i) by (2) and using (4), wc have
dpg _ ^tcGM T'^'^
(ipr KoL P
The equations of equilibrium, then, arc
k 3 d{pT) _ ^ttcGM
pH a p
and
a ^/( 7'0 _ P^
3 dr ^TTcr^ ’
dM{r)
dr
4Tr^p .
(4)
is)
( 6 )
( 7 )
( 8 )
The foregoing equations arc reduced by the substitutions
r = ; r = r„r ;
p — PoO" ;
M(r) = , (9)
294
STUDY OF STELLAR STRUCTURE
to the form
„ d{tTT)
d(r 4 ) ~ “7 “ " ’
(10)
and
t 2 _ 0-“
7-3 • s ’
(ll)
provided
(12)
Tl'^ KoL I
Po 4 ’rcGiU ’ p 2 ~ 4irc 0 R ’
(13)
K = — i £1 . n 4 ’rR^Po
pfi-arr ^ M •
(14)
The solutions of equations (13) and (14) arc found to be
. _ / 47 rcGlf \8 /3 GM\?
\ KoL ) [a Ji ) ’
(is)
j, _ ^wcGMy ^3
(16)
„ ^ 47 r.R^ / 4’rcGlfy U Gilfy
■W" \ KoL y Va A y ’
(17)
and
A' = A 3 / 4 fl-cGMy A CM\
p// a\ KoL J \a R ) '
(18)
Numerically, the foregoing arc equivalent to
F. = 6.41 X io'« (i - Xl)-‘i‘ ,
(19)
and
A/is
Po = 2. 26 X 1037 (l _ A' 2 )-«/« ,
(20)
^ = 2.78 X 10’^ ;u-‘(l - _
(21)
STELLAR ENVELOPES AND STARS
29s
In equations (19), (20), and (21), above, we have expressed ikf, L,
and R in solar units.
2. The solution of the equations of equilibrium , — Introduce the
variable y, defined by
yji = Ka . (22)
From the equation defining K (Eq. [14]), it can be verified that y, as
defined above, is precisely the ratio of the gas pressure to the radia-
tion pressure, i.e., /S : (i — jS) in our usual notation. In terms of y,
equation (10) takes the form
d{yT^)
d{r^) - y " ’
(23)
which is equivalent to
^ ~ •
(24)
Introducing the new variable defined by
( 25 )
we have the following dilTerenlial equation for y:
Ixy ^ “ y^y + 0 •
(26)
Instead of (26), consider the more general equation
^xy “ y^y + >
(27)
where 5 is a small positive constant. 'To solve (27) we
method in principle due to Jeans.
Assume a solution of the form
shall adopt a
y = y« + 5 y, +• +
(28)
Inserting the foregoing in (27) and equating the cocfhcicnts of the
powers of 5 , we (ind
•V = yo(yo + i) ; .vy„ = -y,(2y„ + i) ; .
. . . , (29)
296
STUDY OF STELLAR STRUCTURE
or
+ ( 3 „)
Hence,
+ (3.)
The foregoing series converges quite rapidly for small S. Thus, for
the case where S = 1/8, we have
The second term in the brackets in (32) contributes, at most, about
3 per cent. Also, since x is very large, except in the immediate neigh-
borhood of the boundary of the star, it is clear that for our present
purposes it would be sufficient to replace the second term in the
brackets in (32) by its limiting constant value -1/32, in which case
y = Uyo ■ (33)
Reverting to the original variable t, we have, according to equa-
tion (29),
/sTt*/’ = * = y„(y„ + i) , (34)
or
^ yliyo + ly . ( 35 )
By (22), then,
<^ = 2yr^, (36)
or by (33) and (35)
+ I)^ (37)
Equations (35) and (37) determine t and a in terms of yo. We shall
now determine the variation of y„ with
From (10) and (ii) we have
^ d((TT)
^J-dT
(38)
STELLAR ENVELOPES AND STARS
297
or, in terms of y„ (cf. Eqs. [35] and [37]),
— ^
zr
K " yl{yo + lY di
Equation (39) is found to reduce to
[yt(yo + i)*] = - 1 + ^
31 I
32 yo + I ■
or
yo(yo + i)“(i7y. + 9)dyo = -k j.
(39)
(40)
(41)
We shall presently see that, except in the very immediate neighbor-
hood of the boundary, yo ~ 10 (often it is very much larger), so
that we can properly neglect the term 1/32 in comparison with yo in
equation (41). It should be remembered in this connection that in
the immediate neighborhood of the boundary y — > y„ (according to
the solution [32]), so that we make the best of both “worlds” by
neglecting 1/32 in (41).' We therefore have as the (yo, |) differential
equation
(y<j + i)^(i7y., + 9)</y„ = — A' || . (42)
Integrating the foregoing equation and using the boundary condi-
tion that at 5 = I , yo = o, we have
( y « + i>’(5iy.. + 19) - 19 = 12A' . (43)
Equation (43), combined with (35) and (37), determines the physi-
cal structure of the stellar envelope completely.
To obtain the mass in the envelope, we have to integrate (12).
We have
= jT ^ • (44)
‘ It should be noticed that the solution (32) is a singular solution of the dilTerential
equation (27) (cf. chap, ix), to which all its other solutions very rapidly converge, so
that, in any case, we should be careful not to take the behavior of the solutions in the
immediate neighborhood of the boundary too literally.
298 STUDY OF STELLAR STRUCTURE
Using (42) to change the variable of integration to y„, we have
^(i; S) = + 9)<iyo ;
or, finally, using (43), we have
Hyo) — ^
32 A®
yliyo + i)*(i7y° + 9)<^y°
^ {(yo + iK3'o + J?) - ;.?) + i]'
Put
where
w = O3»o = I ? ay ,
II
4K-
a 4 = il
( 4 S)
(46)
(47)
(48)
Equation (46) now reduces to
where
V w'^(w + aYiw + Aa)dw
As stated on page 292, we shall define the extent of the stellar en-
velope by the fraction (i — J*) which contains the outer to per
cent of the total mass of the star. By (9) this means that
^(i;n
I
loD ’
Let w = where J = J*. Then by (49)
(sO
i_ ^ 31
10 32
(52)
or, using the explicit expressions for K and D given in equations
(17) and (18), we have
^ 3£ 411^ /_4 3 GMV’ ” UircGM 'S°-^ <A-
10 52 M \iT a R ) \ KoL ) \ 6 3/
/(a; w*) .
(S 3 )
STELLAR ENVELOPES AND STARS
299
Inserting the numerical values for the various quantities occurring
in (48) and (53) and expressing ikf , L, and K in solar units (which con-
vention we shall adopt hereafter), we have finally the following
equations which determine the central condensation of any star for
which L
, M, and R arc known:
a = 6. 25 X 10“^
IIM^
(54)
/(a; ?c^*) = 0.0618 -
I -
/£/{.. .5 y/»
(SS)
and
^ (w* +
I
+ i?a) +
I -
( 56 )
Equation (56) is obtained from (43), which, in terms of w (= ay«),
has the following form:
(w + a)’(w> + d’l'a) — = (|- - . (57)
3. Stellar envelopes with ncgligihlc radiation pressure. -From (47)
it is clear that a is a measure of the importance of the radiation
pressure (for some typical stars [cf. § 5] a ^ 0.05). We shall con-
sider now the case of negligible radiation pressure, i.e., the case
where a is small.
Let us first consider the case of vanishing radiation pressure.
Then y„ is very large, and a can be neglected in comparison with
unity. According to (35) and (37), we have for the case under con-
sideration
so that
.VI I ..
(Vu ~>
(58)
(y„ — >
co) .
(59)
I2A' (j- l)
(y. ->
(60)
Again from (43)
300
STUDY OF STELLAR STRUCTURE
or from the definition of w (Eqs. [47] and [48])
i + w* =
(y„->oo), (61)
a relation which could have been obtained directly from (57) by
making a tend to zero.
Using (60) to eliminate y„ from the relations (58) and (59), we
have
(y„ -» 00) , (62)
= 31 / 4 I fi _
32 \i7/ ir3 « V ■
Using (52), equation (63) can be reduced to the form
I /i Y’’’’*
io£»/(o; w*) U y ■
Since
p = poo* ; T = Tqt , (65)
we have, according to equations (15), (i6), (17), (18), (62), and (64),
y _ 4 Mfl GJIf /i
17 * R \|
and
__ I _/l \fi.2S
^ 30/(0; TO*) ’ (67)
where p is the mean density of the whole star. If wc put $ = J* in
the foregoing equation and use for / the value given by (55), wc
shall obtain the density and the temperature at the “base” of the
Stellar envelope.
Equations (62) and (63) show that stellat envelopes with ficgligihle
radiation pressure form a homologous family.
For = o, the equation determining the extent of the stellar en-
velope simplifies considerably. From (50) we now have
STELLAR ENVELOPES AND STARS
301
Equation (68) is an elementary integral and can be evaluated. The
result is
log
+ I
2 + I
Il^W^
3(^4 1)3 24(w‘^ + i)^ g6(w^ + i) ■
(69)
For the case under consideration, namely, that of vanishing radia-
tion pressure, $ is related to iv according to equation (6i). In Table
17 the function /(o; is tabulated.
If a is small (but not vanishingly small), we can obtain for /(a;
w) a three-term Taylor expansion in a :
/(.;») =/(o;») + .(£)__^+|j(g), (70)
when terms of higher order in a arc neglected. From the definition
of /(a; w) (Eq. [50]), we verify that
(rO... - <’■>
and
(^\ - so(, t- r
Ua^/a=o~ Jo (™<+ l)-
- 321(8 + A)(3 + J?) + 3(1 +
+
10 I , \ C"
' w^hiw
-f i)s
(72)
The integrals occurring in (71) and (72) arc all elementary and can
be evaluated.
We write (70) in the form:
/(a; w) = /(o; w) -f- aA,/n + a"A/o , (73)
where the explicit expressions for A,/„ and A ./„ are easily found from
equations (70), (71), and (72).
302
STUDY OF STELLAR STRUCTURE
To determine the relation between $ and w to the order of accuracy
we are working, we write
+ <«>
where (cf. Eq. [57])
and
So =
iw + aYiw + JJa) - + i
I
I + *
We re-write (74) in the form
S = So + ctAifo + a^Aafo ,
where it is easily found that
-(3 + i?)
and
+ i)^
(75)
(76)
(77)
(78)
(79)
In Table 17 the values of /(o; 2C>), A,/,„ A/,„ A,$„, and Aji’„ are tabu-
lated with argument $o- This table, combined with equations (73)
TAULK 17
/(o; 0
A,/„
Aifo
A-io
0.90
+0.000014
+0.00020
+0.001 14
-0.5257
— 0.8047
.8s
. 000085
+ .00100
+ .00436
.6634
- .7310
.80
. 0005 2
+ 00305
+ .01166
■ 765 I
— .5807
•75
.00087
+ .00711
-1- ouSys
.«322
- .4138
■70
.00201
+ .01405
+ .02709
.«753
- .2265
.65
.00415
+ .02461
+ 03143
•S057
- .0424
.60
.00789
+ .05925
+ .02586
.8058
+ .1270
■55
.01414
+ .05788
4- .00304
.8776
+ .2738
■SO
.02417
+ .07942
- .04523
.8431
+ .3024
•45
■03991
+ .10125
— .12674
■ 7030
+ .4787
.40
.06421
+ .11815
— . 24684
■7314
+ - 5.^04
•55
. 10148
+ .12068
- .40521
•6572
+ • .5468
■50
.i 5«75
-j- .09222
- .50024
•5730
+ .5.^85
•25
.24771
+ .00342
— . 76900
.4805
+ -4777
0.20
+0.59051
— 0.19096
—0.86837
— 0.3816
+0.5085
STELLAR ENVELOPES AND STARS
303
and (77), will enable us to determine /(a; ?) for small values of a.
Actual comparisons with the values of /(a; $), computed accurately
from the integral which defines it, show that the approximate
solution obtained by using the table is correct to within i per
cent for a ^ o.i ; £or a = 0.15, a maximum error of about 3 per cent
is made.
If a cannot be neglected in comparison with unity, recourse must
be had to numerical methods to evaluate the function /(a; w). For
practical purposes it is convenient to tabulate /(a; w) for different
TABLE 18a
0 ==
o.os
a
= 0.10
a
= 0.15
a
= 0.20
a
= 0.25
X
€
/
/
€
I
/
f
/
0.3s
0.935
0 . 0000002
0.40....
0.948
0.910
0 . OOOOOOf)
.908
000001 '>
0 882
0.45... .
.925
0 . 00000 1
.902
. 000003
. 000005
•«44
.000009
0.50. .. .
.890
. 000007
.868
. 0000 1 2
.836
.000021
.801
. 000034
0. SS
.861
. 00002 5
.828
. 00004 ^
• 7 <)iJ
. 000069
■ 75 ?
. 000105
0.60. .. .
.82 1
. 000084
•7S8
. 000 I 34
•744
.000202
.702
.000287
0.05... .
0.813
0.0002
.775
. 000246
■ 785
. 0003618
.692
.000520
.649
.000700
0.70....
.767
• 0005
, 72 (.
. 00064 1
• 683
.000()02
• ^>86
.001 206
• 56 ^>
.00154
0.75
.718
.0011
.674
.001502
• ^>80
.002000
.586
.002544
•548
• 003 1 1
0.80... .
.066
.0025
.621
.003205
.577
.00406
•584
• 004936
•462
.00578
0.85....
.612
• 0050
.568
.006290
•5-\S
.00762
•4«8
. 008894
•444
.01005
o.go. . . .
■ 550
.0095
.516
. 01146
•475
.01388
•48^’
.01502
■ 866
. 0 I 643
o.()5- ■ . ■
.508
. 0 1 69
.467
•01958
.428
. 02 1 (>4
■861
.02396
•857
.02548
1.00....
■450
.0281
.420
■03136
• .?S4
.034 1 <)
0.350
0.03633
0.816
rr',
0
6
1.05....
.413
.0441
•877
. 04792
0-.M4
0.05082
1 . 10. . . .
• 170
■ 58 1
. of)6 1
• 887
. 06994
1.13
. 0948
■ 80 1
. 0{j8 1 1
1.20....
. 2<)0
.1311
0. 269
0. 1330
1.25
0.264
0.1752
specified values of a. The numerical integration has been effected
for the cases a = 0.05, o.io, 0.15, 0.20, 0.25, 0.50, and i.oo; the
results are tabulated in Tables iSa and i8ft.
4. General remarks. An important quantity which has been iso-
lated is a; this determines the relative importance of the radiation
304
STUDY OF STELLAR STRUCTURE
TABLE m
X
o.S
a =
I.O
/
(
/
0
I .0000
0
0.05
0 . 896
0. 10
0 . 706
0 . 000000002
0.15
0.93s
0.703
. 00000007
0.20
.902
0 . 00000002
0.619
.0000006
0.25
.862
.00000017
0542
. 0000033
0.30
.816
. OOOOOI I
0.47s
.0000126
0-35
.766
.0000052
0.416
.0000375
0.40
. 712
. 0000195
0.364
.0000936
0-45
•657
.0000617
0-319
.0002044
0.50
.601
.0001675
0.280
.0004017
0.5s
•547
.0004011
0.246
.0007253
0.60
•494
. 000863
0.217
.001222
0.65
•44S
. 001694
0.192
.001945
0.70
•399
.003075
0.170
.002952
0-7S
•35<5
.005222
0.I5I
. 004302
0.80
.318
.00837
0.134
.006057
0.8s
. 284
.01276
0.120
.008278
0.90
•253
.01865
0.107
.01 1026
0.9s
.225
.02624
0 . 096
•014359
I . 00
0.201
0.03576
0 . 086
0 . 0 I 834
pressure in the stellar envelope. Let be a point where = i , or,
according to (57),
" (i + a)3(i + n*«) + I -
If o is small, we can write, according to equation (77) and Tabic 17,
= 0.5 — 0.8431a + 0.3924a’ .
By (47) at f = f,, we have
y(f.) = ^ ,
320
or, according to (54),
J?a = 6.4s X 10
r z.’jgM(i - xiy
[ MHi
■ </4
(81)
(82)
(s.O
This is a purely formal definition. It can happen that < f *.
STELLAR ENVELOPES AND STARS
30s
Hence, the particular combination of L, M, and R which occurs on
the right-hand side of the foregoing equation determines whether,
for a particular star, the radiation pressure is important or not. It is
important to notice that a knowledge of all three parameters L, M,
and R is required to determine the relative importance of the radia-
tion pressure. It is therefore satisfactory that for normal stars — i.e.,
ordinary giants and dwarfs — a correspondence is found to exist be-
tween (i — jS) at (determined according to [83]) and (i — j8c)
(determined according to Eddington’s quartic equation). Thus, on
the assumption that /x = i, wc find that for the sun and Capella A
the quantities (i — jS) at are 0.004 and 0.041, respectively, while
the quartic equation yields for (i — jSc) the values 0.003 ^-nd 0.046.
The fact that the observed sets of values for L, Af, and R for the
normal stars predict values for (i — jS) at $ = in such close corre-
spondence with the values of (i — ft.) according to the quartic
equation is a confirmation of the adequacy of the standard model
(in its first approximation cf. § 7, chap, vi) for these stars. On the
other hand, we shall see that this correspondence fails when the
very massive Trumplcr stars are considered. For these stars we
should normally expect (i — jS) to be quite near unity, while ob-
servationally the radiation pressure is, in fact, quite negligible in
the envelopes of these stars; wc have here, therefore, a breakdown
of the theory which has been found to be applicable to stars of ordi-
nary mass. We shall return to these questions in § 6 (cf. Table 21).
Let us now consider stellar envelopes with negligible radiation
pressure, i.e., stars for which a o. The equation determining the
central condensation of the star can be written as (Eq. [55])
/(o; ^*) = 0.0618
( 1 ~
(84)
'Fhe occurrence of M^'^) on the right-hand side of (84) is
easily understood. For, according to the discussion in chapter vi,
§ 6, stars with negligible radiation pressure form a homologous fami-
ly, and, further,
(85)
3o6 study of stellar STRUCTURE
is a homology invariant. Also, we have already shown in § 3 that
stellar envelopes with negligible radiation pressure form a homolo-
gous family. Hence, is also a homology invariant; and, as it de-
pends on L, M, and we should have
I* = function [(^^s)] • (86)
Equation (84) is simply the explicit expression of this form of de-
pendence.
Another feature of (84) which should be noticed is its remarkable
similarity to the luminosity formula (Eq. [14], vii) for the case
18 I. By equations (14) and (107) of the preceding chapter we
have
L
o. 184
I
MSS
(i - X2) i?-s
(87)
It will be remembered that 2 , which occurs in (87), is a certain har-
monic mean value of t taken through the star (cf. Eq. [8], vii); it is
accordingly different from which occurs in (84). Equation (87)
can be re-written in the form
(i -
Ms s j 10-5^3 ’75
0.429 .
Comparing (84) and (88), we have
/(o; ?*) = 0.0618 X 0.429 = 0.0265 ;
( 88 )
(89)
or, interpolating among the values of /(o; in Table 17, we find that
?* = 0.496. (90)
Now the model specifically underlying the luminosity formula (87)
is the standard model with a density distribution corresponding to
the polytrope n = 3. An examination of the Lane-himden function
d, shows that the polytrope n = 3 has a central condensation of
approximately 0.504. The agreement of J* = 0.496 with the “theo-
retical” central condensation of 0.504 proves the consistency of the
STELLAR ENVELOPES AND STARS
307
model for the stars of negligible radiation pressure ; the consistency
here proved may be compared with the discussion of the assumption
= constant” in chapter vi, § 7.
5. Central condensations of some typical normal stars: dependence
on chemical composition . — ^We shall now proceed to apply the theory
we have developed to derive the central condensations of some typi-
cal stars. The data on the masses, luminosities, and radii of the stars
has been supplied to the writer by Dr. Kuiper, who has undertaken
a critical re-examination and rediscussion of the relevant observa-
tional material. It is beyond the scope of the present monograph to
include Kuiper’s discussion; such discussions should, however, be
regarded as an integral part of the study of stellar structure. For
the derivation of the data used here, reference is made to Kuiper’s
investigation in the Astrophysical Journal, 88, 472, 1938. (It should
be pointed out that the absolute bolometric magnitude of the sun
which Kuiper adopts is -4-4.63.)
a) Capella A . — To illustrate the method of calculating the cen-
tral condensations of stars we shall first consider the case of Capel-
la A. This star presents an exceptionally ‘‘pure” case, in so far as
a preliminary examination shows that the guillotine factor Z,. equals
unity.-'* (Cf. Table 14a, chap. vii).
For the case of Capella A we have
Log Z = 2.08 ; Log M = 0.62 ; Log 7 ? = 1.20. (91)
Substituting the foregoing values in equations (54) and (55) and
putting te = I, we find that
a o.0469[iLi(i - (92)
and
/ = 0.0268(1 - . (93)
To evaluate $*, we shall have to make some assumption concerning
jLt and A"o. To examine first the nature of the dependence of on
ju and A^o, it is suilicient to use the “first approximation” considered
^ This is, in fact, a ^oncral characteristic of the normal j'iants, subf;iants, and M su-
pergiants.
3o8
STUDY OF STELLAR STRUCTURE
in § 3 of chapter vii, according to which )u and Xo are related by
(Eq. [26], vii)
2
I + 3^0 ‘
(94)
Equations (92), (93), and (94) and the tables of the function
/(a; J) are suflhcient to determine as a function of Xq. Table 19
shows the result of the computations.
TABLE 19
The Central Condensation of Capella A
Xo
a
/
(*
0
0.056
0.00199
0.678
.2
.049
.01X4
.542
■ 4
.042
■0351
.441
.6
•03s
.0757
•363
.8
.025
.1x8
.321
.9
. .018
.117
.325
0-9S
0,012
0.098
0-347
In Figure 29 the corresponding (f *, Xo) curve is drawn. The fol-
lowing two important features of the (f *, A''o) curve should be noted;
they are, as we shall see, quite general for normal giants and
dwarfs: (a) The quantity as a function of A"„ has a minimum;
(b) (Xo) intersects the line = 0.5 at two points, one of which
corresponds to the extreme abundance of hydrogen.
An immediate consequence, then, of the theory of stellar envelopes
is the prediction for normal stars of a minimum possible value for
Thus, Capella A cannot be centrally condensed to a degree greater
than that corresponding to = 0.32. The existence of the mini-
mum is easily understood :
If the radiation pressure is negligible, the minimum value of
corresponds to the maximum value of /, or, according to (55) and
(94), to the maximum of
(95)
(i +3X0)3-75.
STELLAR ENVELOPES AND STARS
309
It is easily found that the maximum of (95) is attained for
^ \/25^2 ^ = 0.84; At = 0.561. (96)
For Capella A, a is certainly not “vanishing”; yet the maximum
is not appreciably shifted from the value given by (96). However,
if a is sufficiently large, it can happen that the increase of a with fi is
sufficient to compensate for the decrease in f so that the (^*, Xo)
curve shows only a very shallow minimum, or even no minimum at
all (cf. the case of HD 1337, the infrared component of € Aurigae,
and the M-component of VV Cephei considered in § 7).
Finally, if the values of Xo and ju derived by Stromgren (ai = 1.04,
Xo = 0.29) are adopted, it is found that = 0.486; this confirms
the model underlying Stromgren’s theory. We now see that the two
intersections of the (^*, Xo) curve with the line ^ 0.5 precisely
correspond to the two solutions for the hydrogen content discussed
in § 5 of chapter vii.
b) The sun . — According to (54) and (55), we have
and
a = 0.00625
[mCi -
/=
0.0618
(i - »
(97)
(98)
We see that in this case a is quite negligible for the possible range of
At and A'o. To calculate we shall adopt Streimgren’s values of [jl
and A"„, namely, At = 0.98 and A'„ = 0.37 Ccf. Table 146, chap. vii).
In the case of the sun, the guillotine factor is not entirely negligible,
and to estimate I,, we proceed as follows: On the standard model at
the base of the envelope 7 "* = 0.3T.., and for the sun we find
T* = 6 X To^ degrees. From Table 13 we see that here t = 2.
Since this represents the maximum value of /, we may choose the
mean value of t,. to be about 1.5. The maximum value of U can be
taken to be 2. Using Strdmgren’s value of At and A'o, we find that for
= l, 1.5, 2-0, I
I* = 0.40, 0.42 , 0.44 . j
(99)
310
STUDY OF STELLAR STRUCTURE
We see that the uncertainty in the guillotine factor U does not in-
troduce any substantial uncertainty in the derived values of J*; this
arises from the circumstance that L occurs in the square root. Tak-
ing the case h = 1.5 as typical, we have, according to equations (66)
and (67), the following values for the density and temperature at
the base of the envelope :
T* = 7 .3 X 10^ degrees ; p = i . 8po =2.54 grams cm“^ . (100)
c) f Herculis A. — As we have seen in § 6 of chapter vii, f Her-
culis is considered in Stromgren’s theory to be poorer in hydrogen
than the sun. We shall see that we can confirm this conclusion inde-
pendently. According to Kuiper’s discussion of the star.
Log M = —0.02 ; Log 1^ = 0.596; Log = 0.28. (loi)
Since the radius is about twice that of the sun, the guillotine factor
can be put equal to unity. Using the foregoing values for L, Af , and
R, we find that
a = o.ois2[iu(i - xiyy^‘^ (102)
and
/ = o. 164(1 - . (103)
Comparing (102) and (103) with (97) and (98), we infer (a) that
for equal values of fi and Xo the envelope of f Herculis has a larger
radiation pressure than has the solar envelope, although the two
stars have about the same mass; this arises from the circumstance
that f Herculis has a much larger radius; (b) if the sun and f Her-
culis were characterized by the same values of ix and A"o, then f Her-
culis would have a value for much less than that for the sun. But
we have already seen that stars with negligible radiation pressure
should be homologous. Hence, f Herculis must be characterized by
a larger value of /x, in order that its may be (approximate-
ly) equal to that for the sun; this confirms the conclusion based
on Strdmgren’s theory. Actually, Stromgren finds for f Herculis,
jLt = 1.4s and Xo = o.ii. Using these values in (102) and (103), we
STELLAR ENVELOPES AND STARS
311
find that = 0.44, which makes it, in fact, approximately homol-
ogous with the sun.'*
Kk;. 27
d) 7 ) CaSssiopeiac A. 'This star presents a case somewhat similar
to J* Hcrciilis A. For rj Cassiopeiae, wc have, according to Kuiper,
Log M == — o. 14 ; Log L = — o.o8 ; Log R = — o.o8 .
< It should j)L*rhaps be mentioned in this connection that at this stage in the develop-
ment of the theory it would be unwise to stress the slight “discrepancies”-- the differ-
ence, for instance, between = 0.44 and = 0.42, just noticed. It would Jirst be
necessary to examine carefully the state of ionization in the stellar envelope. It is
clear, however, that with further refinements the theory of stellar envelopes is capable
of including finer details than are considered in this chapter.
312
STUDY OF STELLAR STRUCTURE
Using the foregoing values, we find that
and
■/x(i -
a = 0.007 — — jT-^
- _
f = 0.131
(i -
^3.7S;j.S •
With fj.= 1.25 and Xo = 0.20 it is found that
I , ie “ 2 ,
= 0.42 ; I* = 0.45.
Since we should expect ?? Cassiopeiae to be homologous with the
other stars we have considered so far, we infer that this star must be
characterized by a value of the mean molecular weight somewhat
less than that of f Herculis A but definitely greater than that of the
sun.
e) A star in the Hyades cluster {ADS 3475). — In chapter vii we
referred to Kuiper’s discovery of the relatively poor hydrogen con-
tent of the stars in the Hyades. We shall consider one of the stars
for which Kuipcr has derived values for L, M, and R. ADS 3475
presents a case very similar to that of f Herculis. With Strdmgren’s
values for n and A"„ for this star (jn = 1.42, .V,, = 0.15) we find that
= 0.42 or 0.44, according as h. is taken to be i or 1.5.
/) Sirius A.- As a final example of a “typical” normal star we
shall consider Sirius A. We have for this star
LogL = 1.59 ; Log Af = 0.37 ; LogA = 0.25 . (104)
With Strdmgren’s values of and X„ (n = 0.95, A'„ = 0.36) we
find that
= I ; ^<1=1-5; 4=2, 1
f* = 0.42 ; I* = 0.44; ?* = o.46.[
The (?*, A"„) curves are shown in Figure 27 for the stars considered
in b to f above.
STELLAR ENVELOPES AND STARS
313
6. The structure of the Trumpler stars . — So far we have considered
only normal stars, and the theory of stellar envelopes essentially
confirms the theory described in chapters vi and vii. As an extreme
on the other side, we shall consider the very massive Trumpler stars,
for which the usual theory seems to break down completely. Table
20 contains the data for the Trumpler stars as revised by Kuiper on
the basis of his temperature scale, which should be more reliable
than that originally adopted by 'I'rumpler. These stars occur in the
region of the M R diagram (see Eig. 2), where one would expect
TAH1.K 20*
The Trumpler Stars
Star
T.OK M
Lota L
Lok R
Ti
f .74
5 .«S
0.64
7 a
1 .90
4.72
0.86
7 ’.,
2.14
5 • ^>o
0.78
T.i
2.45
S . 7 ^>
i . 26
7 s
2 .. 35
T.lS
n
2 .(>0
5 04
1 . 22
Ti
1 . 8()
4<)2
O.(>0
* The stars are numlMreil V'l ,7', in Hie order in which
they an* o»ntaiiU!<l in 'rrumiiliT's 'lalilc 1 1 1 {Ptih. A ..S'./*., 47, 25.1, kj.ls).
the hydrogen content to be fairly high, from an exlrai^olation of the
Strdmgrcn curves of constant A'o (established in the region of the
normal stars), 'bhe calculation for the central condensations of these
stars has been carried out for two values of A',, (A'„ = 0.05 and 0.60).
The results arc summarized in 'rable 21. It is at once clear that the
theory applicable to the normal stars breaks down for these objects.
In the calculations, electron scattering has been neglected; but
it is clear, from the empirical rule stated on page 272, that we can
take this into account by allowing the guillotine factor to be less
than unity. This would cause the ^*’s (for A',, == o.g.s) given in
Table 21 to lie between the tabulated values of for A'o = 0.95
and A'o = 0.60; for the case A'o = 0.6, electron scattering is seen to
be negligible.
What is, ])erhaps, most striking is the systematic increase of
with increasing mass. The conclusion, then, is that the Trumpler
314
STUDY OF STELLAR STRUCTURE
stars are more or less homogeneous gaseous configurations. This con-
clusion, it should be pointed out, is an almost immediate inference
from the observations. We encounter the “breakdown” nature of
the Trumpler stars also when we attempt to calculate their hydrogen
content by the Stromgren method. As Beer and Chandrasekhar
showed, the problem has no solution.
7. Further applications . — In the last two sections we considered,
on the one hand, the normal giants, subgiants, and dwarfs (for which
the standard model was seen to be a sufficiently good approxima-
TABLE 21
Central Condensations for the Trumpler Stars
No.
Mass/O
Xo = O.QS
Xo = o.6o
a
/
f*
a
/
T,
400
O.OI
0 . OOOOI I
0.90
0.04
0 . 000008
0.90
280
.04
. 000066
• 8 s
. 10
. 00005 ^
.84
Ts
220
■03
. 000073
.85
.07
.000057
.84
Ti
140
.04
. 00029
.79
. H
.00023
.78
T2
100
.02
. 00028
.80
.06
.00022
■79
T,
80
•03
.00071
.75
. 10
•00055
.73
Ti
55
0. 10
0 . 0047
0.59
0. 29
0 . 0036
0.51
tion), and, on the other, the Trumpler stars, for which the model cer-
tainly breaks down. We shall now consider some intermediate cases.
a) VV Cephei: the B-component. — The system of VV Cephei is a
spectroscopic binary which Gaposchkin discovered to be an eclips-
ing system with a period of about twenty years. The brighter com-
ponent is an M supergiant, while the fainter is a B star. 'I’he obser-
vational data for this system are of a provisional character, the chief
uncertainty being in the mass ratio. A value of about i .6 appears to
be the best estimate. It may, however, be as high as 2.2. We shall
first consider the B star and return to the M star later.
For the B star we have, according to Kuiper,
M = 31; R = 28; Log L = 4.22 . (iot>)
STELLAR ENVELOPES AND STARS
315
The foregoing values correspond to an assumed value of 1.6 for the
mass ratio. If the higher value of 2.2 is adopted, we have
M = 46; R = 3S; Log L = 4.40 . (107)
The guillotine factor is found to be unity, and we find that for
X = o. 0.2, 0.4, 0.6, 0.8, 0.9, 0.9s,
f 0.78, 0.71, 0.64, 0.60, 0.58, 0-59, 0.61,
t* = j
I0.82, 0.76, 0.70, 0.67, 0.65, 0.66, 0.68.
The first set of values for |* arc derived on the basis of (106), while
the second set corresponds to (107). The B star is thus seen to
be similar to the Trumpler stars. This is what we should have ex-
pected from the mass and the radius of this star as compared to the
Trumpler stars.
b) V Pup pis. 'I’he two components of this system are nearly
identical, so that for the present purpose it is sufficient to consider
the average of the two components. We have M = j8.6, R = 6.8,
and Log L = 3.87. The ($*, A'„) curve is shown in Figure 28. We
infer from this curve that V I’uppis is probably somewhat more ho-
mogeneous than the less massive stars.
c) Hi Scorpii. 'Phis star has been recently investigated by
Elvey and Riidnick; a rereduction of their claja by Kuiper leads to
M = 12.0, R = 5.50, and Log L = 3.37. 'Phe (^*, A'.,) curve is shown
in Figure 28.
d) The B-com ponent of ^ Aurij'ac. 'Phe system f Aurigae is of
the first class. 'Phe observations by Outhnick and his collaborators
and by Christie and Wilson have been rereduced by Kuiper. We
shall first consider the b-component. P'or this star we have, accord-
ing to Kuiix'r,
yt/ = 8.i; R = $.1 ; Log L = 3.01 . (109)
'Phis star has a mass less than that of the other massive stars we
have considered so far. 'Phe (^*, A'„) curve is shown in Figure 28.
It will be seen that the 'Prumpler stars, VV Cephei (B star), V
Puppis, Hi Scorpii, f Aurigae (H star), Sirius A, and the sun represent
3i6
STUDY OF STELLAR STRUCTURE
a sequence along which the Xa) curves change continuously;
this strongly suggests that the breakdown of the standard model for
Fk;. 28
stars on the main scries sets in at about M = loG, becoming more
and more pronounced on passing toward the larger masses.
e) The K-component of f Aurigae.— Tox this star we have
M = 14.8;
R = 200;
Log L = 3.80.
(no)
STELLAR ENVELOPES AND STARS
317
The ($*, A’o) curve is shown in Figure 29. We see that, though this
star has a mass about equal to that of ja. Scorpii, it seems to have
a normal density distribution. Indeed, with Stromgren’s value of
M and Xo (li = 1.07, X„ = 0.34) we find that
0=0.24; /= 0.0082; $* = 0,466; (ill)
this confirms the standard model for the interior of this star.® The
K star here considered is, in fact, just as “pure” a case as Capella A;
it is a first-class determination, and the guillotine factor is unity.
We have here a suggestion that the massive stars in the supergiant
region of the Hertzsprung-Russcll diagram are probably different
from the equally massive stars forming the extension of the main
series. Though the two cases considered below are somewhat un-
certain from the observational side, they seem to lend support to
the suggestion.
/) The infrared component of < Aurigac. 'rhe data for e Aurigac
have been derived partly by an indirect method and are less reliable
than for most of the other stars we have considered. 'I’hc mass and
the radius seem to be fairly well determined according to the in-
vestigation of Kuiper, Struve, and Striinigren.* 'I’he luminosity is
only appro.ximately known through the recent measures by Hall,
which have been discussed by Kuiper.'' We have
.^7 = 24.6 ; A = 2140 ; J.,og L = 4.46 . (112)
We lind that for
A'., = o, 0.2, 0.4, o.o, 0.8, 0.9, \
o (
i 0.28, 0.2,^, 0.22, 0.24, 0.29, I
respectively. It is Unis seen that the ($*, .V„) curve for this star (cf.
Fig. 29) shows a very shallow minimum ; further, for a wide range in
.V,„ does not dilTer appreciably from the value 0.23. We can there-
fore conclude that the infrared comj>()nent of e Aurigae is probably
■5 Since a = 0.24, the nuliution pressure is (|iiil.c jippreciiililo; as such, the envelope
f Aurigae (K star) is not strictly homoloj>;ous with that of a star having nef?ligil)le
radiation jiressure.
86, 570, j(;,57.
^ //m/., 87, 201;,
STUDY OF STELLAR STRUCTURE
318
much more centrally condensed than the normal stars. This result
is easily understood. According to our definition of a (Eq. [54]), a
large R (and/or L) implies that the radiation pressure is important;
this has the effect of forming an extended stellar envelope for the
star.
A case similar to e Aurigae is presented by the M-component of
VV Cephei.
g) The M^-component of VV Cephei , — As already indicated in (a),
above, the data for this system are of a provisional character. For the
brighter component of the system (which is an M supergiant) we
have
Af = 49 ;
R = 2130 ;
Log Z, = 5.62 ,
(114)
or
M = 102 ;
R = 2630 ;
Log Z = 5.80 ,
(iiS)
according as the adopted mass ratio is 1.6 or 2.2. Computing for
different values of A^oj we find that for
Xo = 0.4, 0.6, 0.8, 0.9, 0.9s,
f 0.065, 0.072, 0.12, 0.20, 0.29, (116)
I0.21, 0.26, 0.32,0.41,0.50.
The first set of values for arc derived on the basis of ( 1 14), while
the second set correspond to (115).
Wc thus sec that in spite of the uncertainty in the observational
material we can conclude that the M-component of VV Cephei must
be highly centrally condensed. Indeed, the possibility that 90 i)er
cent of its mass is concentrated within 5 per cent of its radius cannot
be overlooked. The system of VV Cephei is therefore of quite unusu-
al interest from the theoretical viewpoint, inasmuch as we have in-
dications that the standard model breaks down in the opposite di-
rections for the two components.
Considering, now, the stars in the sequence the sun, f Herculis,
Capella, f Aurigae (K star), e Aurigae (I star), and VV Cephei
(M star), wc infer the possibility of a breakdown of the standard
model also in the region of the massive supergiants (stars of high
luminosity and large radius). The breakdown is now, however, in
STELLAR ENVELOPES AND STARS
319
the sense of becoming more centrally condensed ; this differs from the
case of the massive stars which form an extension of the main series;
the latter are certainly more homogeneous than the normal stars.
h) AO Cassiopchie and 2g Canis Afajoris. Wc shall iinally con-
sider “ovcrluminous” stars, of which AO Cassiopciac is an example.
In this case we have
M = 40 ; /e = 19 ; Log A = 5.77 . (117)
'This star is therefore almost as luminous as the most massive of the
'rrumpler stars. Hecause of this high luminosity with respect to its
mass, we should expect AO Oassiopeiae to be more centrally con-
densed than the massive stars in the sequence fii Scorpii, V Puppis,
320
STUDY OF STELLAR STRUCTURE
W Cephei (B star), and the Trumpler stars. The (§*, A'„) curve for
this star (see Fig. 29) confirms this expectation.
A star quite similar to AO Cassiopeiae is 29 Canis Majoris A.
For 29 Canis Majoris A we have
M = 46 ; R = 20 ; Log L = 5.84 . (118)
Computing for different values of A'o we find that for
^"0 = 0, 0.2, 0.4, 0.6, 0.8, 0.9, 0.95, 1
f ( 1 1 0)
= 0.44, 0.38, 0.34, 0.33, 0.38, 0.43, 0.49. )
The (^*, Xo) curve for 29 Canis Majoris A is therefore (piile similar ■
to that for AO Cassiopeiae. Both of these stars are prohahly some-
what more centrally condensed than even the normal stars.
i) Y Cygni . — A case intermediate to those considered in (//) abovt*
is provided by the system, Y Cygni. The two comi)onents of this
system are nearly identical, and, taking their average, we have
Af = 17, J? = 5.9, and Log L = 4.51. Thus, Y Cygni, though it has
a mass less than V Puppis, is yet very much more luminous. We ejin
therefore expect Y Cygni to be more centrally condensed than
V Puppis. The calculation of for different values of .V„ eonlirms
this conclusion. Y Cygni has probably a “normal” density (list rihu-
tion.
8. Coficludifig TCfH(iTks,~' ihe main results of the loregoing di.s--
cussion can be summarized as follows;
a) The general way in which the theory of stellar envelo|)es sup-
ports the essential conclusions reached in chapter vii eoneerning th(“
structures and the hydrogen contents of the normal stars like the
sun, Sirius, f Herculis, and Capella;
i) ffhe increasing homogeneity of the massive stars on the niain
series, the breakdown of the standard model setting in prohahl\- at
values of the mass of about 10 O ;
c) The centrally condensed nature of the massive siijuTgiaiits
We may also infer from the examples discussi'd that a certain
systematic variation of the stellar model in the (A/, R) plane e.xists.
Only more extended observations will show whether this is a legiti-
mate generalization.
STELLAR ENVELOPES AND STARS
321
BIBLIOGRAPHICAL NOTES
Stellar envelopes with negligible radiation pressure have been consid-
ered by —
1. A. S. Eddington, M.N., 91, 109, 1930.
2. B. Stromgren, Zs.f. Ap., 2, 345, 1931.
A treatment which takes into account the radiation pressure exactly was
first given by —
3. S. ClfANDRASEKHAR, M .N 96, 647, IQ36.
'fhis chapter represents a hitherto unpublished investigation by the author,
except in so far as it overlaps his earlier paper (ref. 3).
§§ I and 2. — Reference 3. The analysis is carried through somewhat more
rigorously than was necessary for the purposes of reference 3. 'Fhe treatment of
the differential equation (27) is similar to a method used by Jeans.
4. J. H. Jeans, ilf.A., 85, 201, 1925. Also, Astronomy and Cosfnogony,
§§ 78-86, Cambridge, England. 1929,
The discussion contained in the rest of the chapter is published here for the
first time. 'Fhe writer is indebted to Dr. Kuiper for providing the observational
material. Some further references may be noted:
5. G. P. Kuiper, A p. J ., 86, 166, i037» where L, M, and R values for certain
members of the Hyades are given.
6. R. J. 'Frumpler, Pub. A.S.P.. 47, 254, 1035, where 'rriimjder’s massive
stars arc described.
7. H. Vogt, A.N., 261, 73, 1936.
8. E. A. Milne, M.N., 90, 17 , 1029 .
9. A. Beer and S. C'iiandrasekiiar, Ohsvrvatory , 59, 16S, 1036.
10. G. 1*. Kuipeu, O. Struve, and B. Stri'ivkiren, Ap. 86, 570, 1037.
'Fhis paper contains the interprelnlion of e Aurigae.
CHAPTER IX
STELLAR MODELS
In this chapter we shall consider some classes of stellar models.
The interest in some of them may be of a rather formal character,
but for the future development of the theory of gaseous stars the
properties of the models considered may afford some guidance. A
variety of stellar models has been investigated by several writers,
of which only a limited number will be included in this chapter.
I. The model e = constant . — This model corresponds to a uniform
distribution of the sources of energy; as such, it represents one limit-
ing case of the possible stellar models, the other limiting case being
the “point-source model” (cf. §§3 and 4).
For the model t = constant we have
, - L(y) _ L
M{r) M ■
(i)
We shall assume for the law of opacity
K = KopT~^'^ .
(2)
The equations of equilibrium are, as usual.
d ( k rr t X rn,\ GM{r)
(.^)
y CJor<) ,
dr 4Trcr^
and
( 4 )
= 47rr"p .
(5)
From (i), (3), and (4) we have an alternative form for one of the
equations of equilibrium :
± 3 l(p7') ^ T’ s
m// a rfCri) k,L p ^ ^
322
STELLAR MODELS
323
y = ^ ^ (7)
I - /3 iill a
Equation (6) can now be written as
d{yT^) __ \TrcGM k 3
diT^) ~ k„L fill a 'Y~
or
AttcGM k 3^,/, , , \
df = ui -a - y^y + • (9)
Introduce the new variable x, defined by
^ 4If.6W ±_ 3 .
k„L fill a ■
We then have, instead of (9),
^ = -v - y(y + i) , (ii)
an equation identical with the one we have discussed in the last
chapter (Eq. [26], viii). 'I’hc solution is accordingly given by (Eqs.
[29] and [33], viii)
^ = y«iyo + i) ; yn = iriy • (12)
By (?)> (>o). and (11) we have
A a 7V. = 32 (,,A\
k„L fill a 31 (i — py \ 31/ '
Eliminating T between (7) and f 13), we (ind
4Trc(lM / k 3y''* _ 32
k„L \fill a )
31 (i - /3)'
I + -
DilTerentiating the foregoing expression, we find that
(i + ^) + J +
,, /3(i - |3) I
31 + J /3(r - /3)
324 STUDY OF STELLAR STRUCTURE
On the other hand, we have
w)
from which we derive
dP 1 { dp 4 — , I
“3 r7“^i - • (^7)
Eliminating between (15) and (17), we obtain
^ = i [4 4 - 3/3 1 dp .
■ -P 3 1 6(1 + ,8) + I + 6)S(i - ;8)(3 i + (3)-'j7’
which can be written in the form
where the effective polytropic index ficn is given by
= (7 + 6^)(3i + g) + 68(i - g)
(i + 3^)(3i + P) + 2|8(i - 8) ■
(19)
(20)
From (20) it follows that for the cases jS = j and /3 = o we have
»ffr= 3 - 2 S (/3 = i) ; n,n =7 (|3 = o) . (21)
Since, in general, we are interested in values of (i - /3) ~ o.oi; (or
less), it would be sufficient to consider the case of ( i - ~ o.
Then it is a sufficiently close approximation to regard the configu-
rations as polytropes of index n = 3.25 (which is constant through-
out the configuration), and therefore is described completely by the
Lane-Emden function :
p = ; T = ; p = . (22)
We may note the form which the mass relation takes. Since there
is a relation of the form
P =
(23)
STELLAR MODELS
32s
where K is a constant, we have, on comparing (16) and (23) at
P = Pc,
K =
3 I -
a J
pV^^ .
(24)
Inserting the foregoing value of K in the mass relation (Eq. [69], iv)
and putting « = 3.25, we have
fit .1
(25)
This is a quartic equation for (i — dr)- Comparing (25) with the
corresponding equation for the standard model, we have, in an ob-
vious notation.
where
A/ (3.2s; I - dr) = A/( 3 : I - dr)-^/,
Jm =
4 /
(26)
(27)
From the constants of the Lane-Kmden functions given in 'rable4,
chapter iv (p. 96) we find that
7,1/ = 1 .0581 . (28)
In a manner quite analogous to the standard model (chap, vi, § 4),
similar formulae for the physical characteristics may be found. We
shall consider here e.\-])licitly only the luminosity formula
47rr(/A/(r — ( 3 ,.)
(29)
which can be written in the form (cf. chaj). vi, § 7, lirst a])proxima-
tion)
4 .Trc(iM{i — p,.)
KeVr. ’
( 30 )
( 31 )
where
STUDY OF STELLAR STRUCTURE
Equation (30) is, of course, an exact equation. According to (2)
and (22), we have, for the model under consideration.
or, as is easily found,
(32)
z - '^•^5
4 •
(33)
Since (cf. Eqs. [60] and [78], vi)
k [ — 4.25 ?^3.2s]f-ii«?3.as) ^
V = If £ 'r-1/2
3 I - 13/ “ ’
we can re-write the luminosity formula as
j, 4TrcGM 1 3 f k
On the other hand, the quartic equation (25) can be written in the
form
(^) 5 w..,. to)
Eliminating (i — / 3 „) between (36) and (37) and using (33), we find
that
tW 4 V/GViV-^ac
4’ * U.2sj U j 3 «o
Comparing (38) with the luminosity formula on the standard model
(Eq. [142], vi), we can re-write the former as
iL (^'y ^ ^ I
43s \ ^ / 3 /c..7,.(3.2S
(30)
where
W 3 .a 5 ) - MV ’
STELLAR MODELS
327
Numerically,
^c( 3 - 2 S) = 1.251 .
(41)
But, according to the first approximation considered in § 7 of chap-
ter vi, for the model e = constant, we found 77c = 1.14 (Eq. [143]).
It is thus seen that the first approximation in the perturbation theory
considered in chapter vi leads to a luminosity formula practically
identical with the one just obtained by a rigorous method. It should
be remembered in this connection that, since the quartic equations
which determine differ by the factor Jm (Eq. [27]), the ft in (39)
is somewhat larger thzxv the / 3 c derived on the standard model, and as
such the difference between 77^(3) = 1.14 and 77c (3.25) = 1.25 is
almost completely compensated.
2. The models 77 « — By definition we have
Ur)
_ dUr) _ M(r)
* dMir) ’ ^ ■
M
(42)
It is therefore clear that in the immediate neighborhood of the cen-
ter, € and 77 become identical, apart from a constant factor. More
precisely,
(«)
or, if € is the average rate of liberation of energy for the whole star,
(43) can be expressed in the form
+ <«)
Thus, as r —> o, we have
*=e,+ O(r0; ^ = + ^^5)
We thus sec that in the immediate neighborhood of the center the
assumption
f] oc p'^T" (a ^ o, v ^ o) (46)
becomes equivalent to
e oc p'^T" .
(47)
328
STUDY OF STELLAR STRUCTURE
But the analysis according to (46) is more elementary than that ac-
cording to (47). We shall therefore restrict ourselves to a considera-
tion of the models (46), remembering, however, that the analysis
may also be regarded as a start on the more difficult (and physically,
the more interesting) models (47).* For a law of the form (46), equa-
tion (44) takes the form
_ - r j_ 1 f “ I 1 1
p (0 \p p i J
(48)
For positive density gradients (i.e., dpjdT ^ o), the terms in the
curly brackets in the foregoing expression are negative; it is, there-
fore, clear that the right-hand side of (48) will vanish for some value
of r (say r*). For r greater than r*, equation (48) will give negative
values of e. Consequently, for the models of the type considered,
we should “break off’’ the solution at r* and consider a “point-
source” envelope for r > r*.
We shall now proceed to a discussion of the models (46), for which
wc can write
rj = ; 77., = I . (40)
Further, for the law of opacity we shall assume
K = Kop'^T~^~^ (/i > o) . (50)
From the eciuations of equilibrium (3) and (4), and (40) and (50),
we derive
k 3 d{pT) _ /^ircGM
lxIIad(X*) KoVo P'‘+''
The foregoing equation is reduced by the substitutions
to the form
— PoO" ;
T = r„T ,
( 52 )
II
(5,0
' Al the present lime no systematic investigations of the models (47) exist. A fairly
comi)lcte analysis of the models (46) has been given by Chandrasekhar.
STELLAR MODELS
329
provided
Po a ^ ^ fc..77o
Ti ^ 3 ’ pS^" /^ttcGM ‘
Equation (54) can be expressed alternatively as
po —
r. =
/ ^ fY’'*’' " ( KoV a Y^ | *^‘
A k 3) WcGM) .
K illl /gor7ft .s(i-a-«)+s-/^
k 3/ ^ttcGM
Equation (53) now becomes
or
d((TT)
(It
4T^
I
>
d(T ,
-l-H Hr
+
4T'
■fH-N— _
(S 4 )“
(ss)
(S6)
(5 7 )
(58)
From this equation it follows that two critical cases arise, namely,
when j/ = 3 + 5 and when */ = 6 + '’»• In the former case, we have
~ + 4t’(( 7« ~ i) = o , (59)
(It
from which it follows that alon^ all solutions of the equation (50)
<r I as r — > 00 ^ 4- s) . (60)
Again, \{ v = 6 + s, the (<r, r) dilTerential eciiiation is
' ' ' + 4()r"l''TJ — 4 = 0. (61)
(IT
From equation (61) it follows that for a solution which has no singu-
larity at the origin, a must tend to a linite limit as r — > o:
a — > 4'^^“ *■" ^ (r — > o) . (62)
=* 'I'hcsc transformiilions arc not possible for the case 3 -f .v — = ^(aH-w).
330
STUDY OF STELLAR STRUCTURE
More generally, we find that all solutions of (58) must tend asymp-
totically to a certain singular solution whose behavior at infinity
is determined by an asymptotic series, the dominant term of which
is seen to be
3+J—
a T (r — > 00 ) . (63)
In the same way, the solution which has no singularity at the origin-^
has the following behavior at that point:
4(a -|“ W -f“ l) a-hw+i
Q. S ^ fl — V_
6+.'?— y
(64)
We shall now consider another form of the differential equation (58)
which isolates certain cases for which the integration can be carried
out explicitly. Let
z = err; / = . (65)
Then, instead of (53), we have
dz ;l(3+“+>‘+»“i')
di ~
( 66 )
From the foregoing equation it follows that if
i/ = 3 + a + w + ^, (67)
the equation can be integrated. (The case a + w = o can also be
integrated; but this case is not of much physical interest, as we
should expect a ^ o, w ^ o.) For the case (67) the integrated form
of (66) can be expressed in the form
/ = C -
( 68 )
a Such solutions exist only for i' ^ 6 -f-
STELLAR MODELS
331
where C is a constant of integration. For specified values of o +
the integral occurring in (68) can be evaluated. The foEowing ex-
plicit forms of the solutions may be noted :
/ = C — {z 4 - -h 2 log I (i — z'/-*) I j , a, + n = \ , ]
. _i_ r ('^9)
*' = 3-5 + ^. J
/=C-{z-(-log|(i-z)lj, a-|-«=i, j' = 4-f-s, (70)
i = C - |z - 1 - § log ^|, a + n = 2, ^=5-1-5, (71)
— {
z + G- log
(i - z)'
!" + Z -h I
a H- M = 3 )
. _. 2Z -h I , I
V3 vl
V = 6 + s ,
v;/’
and
/ = C - I z -h J log ^ J - I- tan-‘ z | ,
a + w = 4 , »/ = 7 + .
In the foregoing equations, /( = r^) is proportional to the radiation
pressure and s( = err) is proportional to the gas pressure (the con-
stant of proportionality in each case is the same [cf. Eq. (54)];
is jS : ( r — in the usual notation).
As would be expected from the earlier discussion, for the case
a + n = 3 the solution which has no singularity at the origin tends
to a finite limit as r — > o; for the case under consideration, <t — > \/2
as r o. 'Fhe general nature of the (tr, r) relations for these
models is shown in Figure ,^o. From an examination of this figure
and from the earlier discussion of the (a, j/) models we have the fol-
lowing theorem.
If e cc p"7’" and if, further, i' ^ 6 + 5, then in the immediate
ncijflilwrhood of the cc 7 itcr either the density falls off extremely rabidly
with temperature almost a vertical dro]) of the density with de-
creasing tem])erature'’ or the density gradient is always negative, i.e.,
4 Actually, it is easy to verify that dfr/dr — > co .
332 STUDY OF STELLAR STRUCTURE
dpIdT < o. // y ^ 3 + j, and a + « ^ o, thm we always have (even-
tually) positive density gradients for increasing temperature.
3. The point-source model with k = constant.— In the point-source
model it is assumed that the entire source of energy is liberated at
the center of the star; analytically, the assumption is that L{r) =
constant = L. The point-source model, then, is another limiting
Fio. .30.— (o-, t) variations for the models (k=k„ p* and
The curves i and 6, 2 and 7, 3 and 8, 4 and 0, and
the cases o+w = o.5, 2, 3, and 4, respectively.
V — Vo /)" 7’*') with
5 and 7 0 refer to
case for the possible stellar models; the uniform distribution of the
sources L(r) « M(r) is another.
The point-source model with k = constant presents certain sim-
plifying features, and has been studied by Cowling and von Neu-
mann. Von Neumann s treatment of this problem is verj' powerful
and, as such, is instructive as an example of the application of meth-
ods and principles which should be of quite general value in the dis-
cussion of other stellar models.
STELLAR MODELS
333
The equations of equilibrium for this model are
d{p + Pr) _ GM{r)
dr 7^ ^
(74)
dpr kL
dr 4 Tcr^ ^ *
( 7 S)
we have
dp 4 TrcG . .
dp,- ,L "W--.
(76)
+
11
(77)
DilTcrcntiating (77), we fnul
Again,
kL d'p , dr iCiTT^c
d“p _ _ j
d /i\ _ i dr _ 4Trc ^
u/ r~ dpr ^ ’
jL (1) = 4e; i_
dp, \rj kL till \a) ^ ‘
Equations (70) and (80) are reduced by the substitutions
r = ; p = \\z ; pr = Wrt
(82)
to the forms
d^z
dF=-^^
(83)
and
MO =
(84)
provided
o'UllI;^ = !L 47 r’r 6
/l“ L‘
(8s)
334
. STUDY OF STELLAR STRUCTURE
and
( 86 )
Equations (85) and (86) can be expressed in the alternative forms
and
" - (j^y <**>
The mass relation (77) now becomes
M{r) =
kL
4ircG L
[nn-g + .
]■
(89)
or, according to (88),
"(') - [ t ^. {^)‘
Finally, if we introduce the npw variable x, defined by (“Kelviivs
transformation”)
^ = r* , (91)
the fundamental differential equations are
dx I /T\ d z /wl^
^ = (I);
We shall refer to the foregoing equations as (I) and (II) respec-
tively.
We first notice that equations (I) and (II) admit of a constant of
homology ; if
X, C'^% Cz, Ct , (93)
the differential equations arc unaltered, and consequently we can
use the foregoing transformation for normalizing purposes for in-
stance, to make the boundary of the star correspond to x = i .
The problem now is to solve equations (I) and (II) with appropri-
ate boundary conditions. Before we formulate these conditions, how-
STELLAR MODELS
335
ever, we shall first consider, following von Neumann, the general be-
havior of the solutions of the foregoing system. For this purpose we
shall consider only such solutions as have a physical meaning — i.e.,
as long as
o < ac, z, / < oo ; 2' 7*^ cx> , (94)
where the prime denotes differentiation with respect to the independ-
ent variable t. Let S : x = x(t), z = z{t) be a solution of (I) and
(II), and let the maximum ^-interval in which (94) holds be denoted
by 7 ( 5 ). We shall refer to 7 ( 5 ) as the regularity interval of 5 . Let
the interval be specified by
o ^ a < t < b ^ +<x> . (95)
By definition, at each of the ends a or 6, x, s, or t must become
o or 00 or s' = <» . We shall first examine the conditions at a.
a) Behavior of the solutions at a.- At a, x, s, or t becomes o or 00
or s' == 00 . By definition, t has here the finite limit a ^ o. We shall
first prove the following lemma.
Lemma i . At a the only possibilities arc x^ = o, z = o, a = o;
further, z.' is finite //xa ^ o.
Proof: It is clear that as / — > a, x tends to a finite limit Xa ^ o,
for, according to (I), x increases with t and hence decreases as
t a. On the other hand, according to (II), s' decreases with t and
hence increases as / --> a; hence, as t--> a, s' must tend to a limit
which may be finite or infinite. But if x,, ^ o, then, according to
(II), s" is bounded, and hence s' must tend to a finite limit Za as
t—^a; this, in turn, implies a finite limit, for 3 as / — >a. If,
hov^ever, z!, = 00, then, since z is increasing, it must decrease as
l—>a and z must again tend to a finite limit, z„. This proves the
lemma.
We have now to consider the two cases a 9^ o and a = o, sepa-
rately.
Let a 9^ o. Then by lemma 1, either Xa — o and/or 3„ = o. If
Za ^ o, then = o, and hence,
.V ^ constant (/ — </) (/ — ^ d) ,
s Wo shall adopt the convention that the term “constant” refers to an absolute
positive constant the actual value of which may change from one equation to another.
336 STUDY OF STELLAR STRUCTURE
or, according to (II),
z" ~ — constant (/ — «)“•< ; (97)
or, integrating twice,
z ~ — constant {t — a)~‘ (/ — > w) . (98)
Thus, a — 00 j which is impossible.
If, on the other hand, 5^ o, then (according to the lemma)
Za = o. But if Xa 9 ^ o, thcn z'a is hnite. As a„ = o, a" < o, and,
since a > o (for t> a), we should have z'a > o. Hence, we can
write
a ~ constant {t — a) {f —> </) , (gy)
or, according to (I),
x’ ~ constant (/ - a)-' (/ — > ,/) , (loo)
or
X ~ constant log (/ - </) {i -> a) , (loi )
which is again impossible, as x would then tend to — ■» .
^ Hence, at a o the only possibility is that .r„ = o, a„ = o, and
z'a= «. We shall now examine the behavior of such solutions at
a 9 ^ o. Put
.r ~ A {1 — a)»‘ ; s = B {1 — a)" . (102)
Since zi = <», it is clear that o < n < 1. Also,/// > o. Substituting
(102) in (I) and (II), we obtain
and
mA{t - «)“*-< = «•///} ■(/ _ „) «
n{n - i)B{t - a)"-^ = - - _ „) ^
Equating the coefficients and the exponents of (/ — (/), we luive
m-i=-n-, »-2=-4w,
Solving (105) and (106), we fim that
m = « = I ;
A = .
= (18./)' > . (107)
STELLAR MODELS
337
Hence,
0
00
and
s ( 1 8 i;) .
(109)
P'rom (io8) and (ioq) it follows that
p oc (/ — ay/i oc oc i . (no)
Hence, the models for which T tends to a finite limit as the boundary
is approached extend to infinity, and the law of variation of density
as r — > « is the same as for an isothermal gas sphere (cf. Eq. [439],
iv).
Let a — o. We shall show that x,, ^ o. To prove this, consider
the cases 3,. 5^ o and 3,, = o.
Inrst, assume that s„ 5^ o. If a-„ = o, then (I) implies that
;V constant
{ 1 a = o) ,
(ill)
or, according to ( 11 ),
d“Z
„ ^ — constant
dl^
(/ -> <1 = o) ,
(112)
or
2 ^ constant /“"■*
(/ — > (/ = o) .
(”.3)
Hence, s — > — oo, which is im|)ossi))le. Therefore, at 1 ^ a =
0, x„
is linite and 3,, ^ o; these solutions corresiiond to the density tend-
ing to infinity as the boundary is approached the tem]>eraUire,
however, tends to zero.
Let us next assume that 3„ = o. By an argument which we have
already employed (Lemma 1) under lhe.se circumstances %[, > o.
According to ( 11 ), s' decreases, and hence, if / ^
s >/='>/=;. (/ /,) (114)
Hence, by (1),
•r' ^ (sO V ■>'' ,
■Y ^ 4 (sO
or
(I ^ /,) (its)
(t ^ h) (116)
338
STUDY OF STELLAR STRUCTURE
if we now assume that *0 = 0 and a = o. If in (116) we put i = tt
and take t for we have
a: ^ 4 (2')“'/"'^ .
Using the foregoing inequality in (II), we obtain
/ dz\~* i^% I I
\dt ) dP 4 < / ° ’
or, integrating, we have
I
3
+ ^ log / ^ o ,
(117)
(118)
(119)
which is impossible, since 2' tends to a finite limit. Hcncc, x„ can-
not vanish for t = o. Along such a solution p o, T — » o, as r — > f?.
We can collect the results so far obtained in the following theorem.
// o ^ a < b ^ CO is the regularity interval of a solution of the dif-
ferential equations (/) and (II), then at the end a there are three possi-
bilities:
la) o < a < 00. Then x# = o, = o, z,', = » and the asymptotic
forms of the solution are
"" ~ 1 ^ ~ (i8a)'A-(; - .
2a) a = o. 7 /z„ is finite, then x^ is also finite and z,', < 00 :
O ^ X, I j Za "K ^ f O ^ Zfi 00 ,
3 a) a = o. // Za = o, then x,, is finite and z.', < 00 :
o < a;,, < 00 , 2„ = o , o < 2' < 00 .
h) Behavior of the solutions at b. At h, by delinition, .r or s or /
becomes o or 00, or s' becomes infinite. Here t has a positive limit
o < 6 ^ “t" Since x increases wiith t, it also has a jaositive limit,
o < ar/, ^ -I- 00 . s' decreases with/, and we therefore distinguish the
two cases; A, s' ^ o does occur and B, s' > o.
STELLAR MODELS
339
In case A, z decreases if t is sufficiently large; thus 2' is negative
when ^ > 6. In case B, z always increases. Hence, sls b, z has
a limit Zb, which is necessarily finite in case A and necessarily posi-
tive in case B.
Case A. — At t = b,z ^ o, and hence s' is negative for some value
of If s' = — c < o for t = ti < by then, since s' always decreases,
s' ^ — c loT t Hence, s can at best tend only to — 00 if
CO, Therefore, for this case 6 00 ; hence, b is finite (and
positive) and, as we have already seen, zt has to be finite as well.
Thus at / = J, Sfe = o or ^6 = .
Now, since Xb > o, s" is bounded; and as / 6, s' tends to a finite
limit, sj. It is clear that ^ —c < o. If s/, 9^ o, then according
to (I), r' is bounded and x would tend to a finite limit as / — ► b;
this is impossible since, if zu is finite, Xb must be infinite. Hence, as
/ — > 6, s Sb = o and x Xb = ^ .
Let us now examine more closely the behavior of the solutions as
s o and r — > . Let = — c, (o < c < co). Then,
or, according to (I),
or, again.
s ^ r(6 — /)
A«/4
^ ^ (6 - /)-« ,
/;«/4
A- T - 0 ,
(120)
(l2l)
(122)
which shows that x— > <» as / — > ft. Equations (120) and (122) cor-
respond to the following behavior of p and T as the center is ap-
proached :
pTcx ($->o) (123)
and
Ti - 7’* ex o) . (124)
From (123) and (124) it is clear that this asymptotic behavior cor-
responds to the case of the density falling olT exponentially as $ — > o,
while the temperature very slowly attains its maximum; the central
regions will be practically isothermal.
340
STUDY OF STELLAR STRUCTURE
A second approximation to s can be obtained by a process of itera-
tion. According to (II) and (122), we have
- 0 I“S (125)
or
^ c^Lq, - t) |log (6 - i )|-4 + . . . . (126)
if terms of higher order are omitted; in (126) the constant of integra-
tion has been chosen in a manner such that z' —> —c as t-^b.
From (126) we have
Z ^ cQ) - t) - (b - tY\\og{b - . . . . , (127)
the integration constant again having been chosen in a manner such
that Zh = o. Finally, from (I),
(it c{b — t) z c{h — 0 * )
The right-hand side of (12S) is seen to approach zero as / h (cf.
Eq. [127]). Hence,
[ ^*^4 I
lim a; ^ log (6 - /) = c,. (129)
/->/; C J
exists. Wo can therefore write
l)i/4
log (b — t) + Co+ (i;^o)
c
Case B.- -As s' > o and decreases, a finite b implies a finite s/,;
this, according to (I), would in turn imply a finite Xh, and this is im-
possible. Hence, b has necessarily to be infinite.
It is clear that as / — > 6 = co, s tends to a limit s^, which may
be finite or infinite. Suppose were finite. 'Fhen, as / — > we
should have
;r' ^ or .r ^ /•‘ 5 / 4 . (13 1)
33co
STPXLAR MODELS
341
By (II), then,
(132)
or, integrating twice (rcmemberiirg that s' — > o as / — > »), we have
z ~
(133)
By (I) and (133) wc have
lim ( .1: ^
/->co\ 5-co J
Hence,
exists and is Unite. We can therefore write
.V: = _ /.'i/.l-l- ^
5*^ to
(134)
(135)
(136)
Equations (133) anti (136) correspond (o (he following behavior of
p and T as the center of the conliguration is ajiproached.
.. /i\i/s
p7’ — > constant ; ^ ’
(137)
in other words, along such solutions T — > <» and p oc 7’"' — > o/’
We have linally to examine the case s^ = co. Jf were linite,
then, according to (ll),
2” = — ■' ^ — (-Veu) ■' < o , (138)
or s'— > -00, which contradicts the hypothesis (case B). Hence,
a; — > 00 as s — > 00 and 00 . Let us next examine the way in
which X and s tend to inlinity. Various trials indicate that s
^*11 nniy hu rcaillod tliiit in the discussion of the (a, if) models we have already en-
countered the heliavior p oc 7’-', T -> in the models v = ^ a + n + s and
a -f n ^ o. Aloiiji; any solution (OS), s — or i as / = t » — > co .
342
STUDY OF STELLAR STRUCTURE
increases at a rate very near to the order of increase of t. We shall
therefore try the following behavior:
By (139)
or, according to (II),
z ^ Ct (log ty (C = constant) . (139)
x~^ ^ (log .
t
Hence, n < o, and we have
X ( (log /)(*-")/4 .
Substituting in (I), we have
— =r — (log ,
or
Z 4(“WC)^/4/ (log /)("‘’0/4 ,
Comparing this with (139), we have
— n — \ \ 4(-~ nCy^^ = C .
Solving (144), we find
Hence, finally,
C = —
3X/3 •
a; ^ ^ /'/t (log /)'/•’
^ ^ ^ • (147)
Equations (146) and (147) correspond to the following behavior of
p and T as the center of the configuration is approached :
T (log r )'/3 oc ±
p cc (log J')”'/3
(?-->o), (148)
(?->o). (149)
Hence, along these solutions both p and T ^ .
STELLAR MODELS
343
This completes the discussion of the behavior of the solutions at
the end b of the regularity. We can collect together the results in
the following theorem.
// o ^ a < b ^ GO is the regularity interval of a solution of the
differential equations (/) and (//), then at the end b there are three
possibilities:
ib) o < b < oo. Then xi, , Zh = o, and zl finite and negative.
More precisely,
b^/A
A- ~ — I log (6—01
and
z ~ c(6 - /) - i (6 - OM log (6 - /) I --t +
2b) b = +O0 and z —> tvhich is finite. Then = o. More pre-
cisely,
and
X
1
I 2
.-111) b = uiid z to. llirn x oo . More precisely,
^r/j ('<« ‘y'-'
and
a\!.\
2 t (log /) •/.» .
r) The number of arbitrary parameters. We have now to deter-
mine the number of arbitrary parameters corresponding to each of
the (lilTerent types 2.1, 3., and ri„ 2i„ 31,.
Solutions /,„ 2,1, and j,,.- Solutions of the type 2.1 and 3,1 satisfy
regular initial conditions and are characterized by three (namely,
s„, and So) and two (namely, and s',) parameters, respectively.
However, in the case la the dominant terms have been uniquely
344
STUDY OF STELLAR STRUCTURE
determined ’without any arbitrary constants, and the number of
arbitrary parameters must be determined by a perturbation method.
Let
and
x = x + ^ (ifi^x) (150)
z = z + tf/ (r^<^z) , ( 151 )
•where x and z are solutions of (I) and (II) such that, as < — > a, their
behavior is governed by equations (108) and (109). Substituting
(150) and (151) in (I) and (II) and retaining only the terms of the
first order of smallness, ws obtain
^ i«/4
dt z*
.
w =
As t—>- a, we have, according to (108) and (109),
and
Put
d^\f/
^ ^sTj
(/ — 0) .
<p ^ A{t — a)’" ; ~ B{l — a)'* .
Substituting the foregoing in (153) and (154), wc find
(152)
(153)
(154)
(155)
Am{t - a)-- = B{t - 0)'-^.. (156)
and
n{n - i)B(t - a)"-^ = A(l - u)’"-*/-’ . (157)
Equating the coefiicients and the exponents of (/ — a) in the fore-
going equations, we find
m ^ n = — I
(158)
A ^ _ n(n — 1)35/-’
(159)
and
STELLAR MODELS
34 S
From equations (158) and (159) we derive that
3 «( 3 » - i)( 3 « - 3) == -8 ,
or (as can be verified)
( 3 « + i)( 9 »” — iS» + 8) = o .
Hence,
n = —I or
5 + •^-7
6
(160)
(161)
(162)
On the other hand, <p x and z implies, according to (108),
(109), and (155), that m> i and » > f. Since w - » = -f,
» > f would imply that m> This excludes the case n = -\
in (162). Thus, the only possibilities are
n
m —
^ 6 ^
(163)
Thus, there are two linearly independent solutions of the type la
for any specified a o; these solutions, therefore, are characterized
by three parameters. We have thus proved: Solutions of the type
la, 2a, and 3a are characterized by three, three, and two parameters,
respectively.
Now the differential equations (I) and (II) arc equivalent to a
single differential equation of the third order, and hence the solu-
tions must form a three-parametric family. (In the language of the
theory of sets of points, solutions of the type i,, and 2a form open
domains in the manifold of all solutions. Solutions of the type 3a
form, however, only a two-parametric manifold and hence can con-
tain no “interior points.” The boundary of la and 2# must, therefore,
be 3a.)
Solutions ih, 2i,, and j*. — Solutions of the types i|, and 2b satisfy
regular boundary conditions and are therefore characterized by three
(namely, b, c, and c„) and two (namely, and 7,,) parameters,
respectively, ff'he dominant terms of the solutions of the type 3b
have been uniquely determined, and the number of linearly inde-
346
STUDY OF STELLAR STRUCTURE
pendent solutions belonging to this class must be determined by the
perturbation method, as in case la, above. Write
X = X <p (<p <Kx) (164)
and
z = z + tf/ (f « S) , (165)
where x and s are now solutions whose behavior at infinity is
governed by the equations (146) and (147). I’he differential equa-
tions for <p and ^ are the same as before (Eq. [152]); substituting
for X and s the expressions (146) and (147), vre obtain
(166)
and
^ ^ Oog •
(167)
Put
(p = (log /)'■ ; ^ (log /)" .
(168)
If m, n, r, and J are not all zero, then the leading terms in equations
(166) and (167) arc proportional respectively to
and
(log ly <x (log
n{n — (log /)“ oc /'"-s/i (log .
(169)
(170)
Equations (i6q) and (170) imply
m — i = n — i ;
r = 5 -H § ;
(*71)
(172)
Equations (172) arc inconsistent; hence, according to ( itiq) and
(170), either w = o or m(w - i) = o. 'I'hus, we have the i)ossil)ili-
ties:
<p = A (log ty ;
^ = Bt" (log ty ,
(17.^)
tp = A(”‘ (log /)’• ;
y// = B (log ty ,
(174)
(p = At’" (log /)'■ ;
yp = Bt (log ty .
(17s)
STELLAR MODELS
347
Substituting (173), (174), and (175) successively in (166) and (167)
and equating the coefficients and the exponents of the leading
ternas, we find:
m = 0 ; « = f ;
r — 1 .
' » )
c =r X
^ » J
■38/3
A:B = ,
4 > 4/3 ’
(173')
w = — -f ; « = 0 ;
»•=?,:
= -«
■ A :B — ^
’ 3 «/.^ 4 S /3 7
(174')
m = J- ; 11 = i ;
'• = - ;
s = - J :
■92/3
(17s')
With w, w, r, and ^ defined as in the foregoing equations, tp and xf/
(Eq. [i68]) satisfy the requirements (p « x and \p <K z, where x and z
are defined according to ( 146) and ( 147) ; thus there are three linearly
independent solutions of type We have thus proved: Solutions
of the type ib, 2^, and 31, arc characterized by three^ hvo, and three param-
eters, respectively. (In the language of the theory of sets of points,
the solutions of the type r j, and 31, correspond to open domains in the
manifold of all solutions, and the solutions of the type 2|, form a
closed set containing no interior points. 'The ‘‘border line” of ii, and
31, must therefore correspond to 21,.)
d) Conditions at the boundary of the configuration. So far we have
considered only the behavior of the general solutions of the dilTer-
ential equations (1) and (11) at the ends “t/” and of the regular-
ity interval. It now remains to select such solutions as can describe
a stellar configuration. At the boundary R of the configuration we
require both p and T to vanish.^ In other words, the requirement is
that, when / = o, s == o. From the earlier discussion (case 3.1),
t = o, z = o necessarily imply the existence of the limit xit — > o) > o.
7 If wc require that at the Iioundury of the configuration, T tends to a finite limit Tn
while at the same time p —> o, then the solution must be such that hir a finite / = /„,
= o. Hence, the solution must he of type la. This solution, as we have shown, corre-
sponds to the case w'here the configuration extends to infinity with /> falling olT as r~^—
i.e., in the same way as in the isothermal gas sphere, h^urther, according to (Sg),
M — > 00 . 'J’hese are, really, only formal dilhculties (cf. Cowling’s paper referred to in
the bibliographical Notes at the end of the chapter), and it is safe to use the initial
conditions (i7f>), since the ratio of Y'd to the values of 7'-' occurring in the far interior
is of the order of ( loV ' ~ lo”"; we can, therefore, certainly put /« = o.
348
STUDY OF STELLAR STRUCTURE
Further, from the homology argument (Eq. [93]), we can normalize
the units in such a way that x = i corresponds to the boundary
i = o, X = I , z = o. (176)
To make the solution definite, another boundary condition is
needed. Assume that
2^ = > o (^ = o) . (177)
A solution satisfying the initial conditions (176) and (177) must be-
long to a solution of t3q)e 3a. The problem presented is twofold: (i)
How does the regularity interval (o, 6) depend on 5 ,? (2) For a
specified 5 „ what is the type of the solution which we arc led to at
the end b of the regularity interval? To answer these questions
we proceed as follows:
Begin a solution of type 2b at / = « and continue it backward for
decreasing /. It is easily verified that as 7„ — -f-oo (cf. Eq.
[136]), the solution is of type 2a as t-*o. On the other hand, if
Yo — » — <» , the solution we are led to is of type la- Hence, an inter-
mediate value of Yo which leads to a solution of class 3a must exist.
On the other hand, it is easy to see that as 5 , — >• <» we should eventu-
ally have solutions of type ib- We can therefore conclude that Ihere
exists a value of di = do such that a solution satisfying the boundary
conditions {176) and (177) with 5 i = 5 „ is of tyfe 2,, ax t -> <» . If d,
> da, then the solutions are of type /j; and if 5, < 8„, they are of type
3 b-
e) Boundary conditions at the center: discussion of the point-source
model— Vic must next consider the boundary conditions at the
center. This is a more difficult problem, since we cannot e.x]}ect that
in the point-source model the equations of radiative equilibrium
will be valid right up to the center of the configuration ; the condition
for the stability of the radiative gradient (Eq. [45], vfj will certainly
become invalidated as we approach the center. As has already been
explained (see p. 228), it is a rather delicate matter to continue the
solution beyond the point where the instability of the radiative
gradient sets in. For the present, however, we shall continue to dis-
cuss the point-source model as though the equations of radiative
STELLAR MODELS
349
equilibrium were universally valid, in an attempt toward the enu-
meration of the possible configurations.
i) The complete point-source model . — ^We require that at the center
of the configuration
M{r) = o (r = o) . (178)
By (90) this means that
From (179) it is clear that 8^ is finite. Thus, a configuration which
satisfies the boundary conditions (176), (177), and (178) must be
described by a solution which is of type 3a as i — > o and is of type ib
as t — » lo- By the theorem proved in section d it follows that 5 i > 80.
We can rc-writc (179) as
or, numerically,
= K-‘u~^sy^ X 1.471 X 10'.
Lq
(180)
(181)
'J'hc mass relation (90) can now be written as
M =
I ()7r'*c’(/* \fxll / a
i/.i
( 5 . -t- 8.) ;
(182)
or, eliminating L from (182), we have
= (it,) ' ck i^l)
or, numcTically,
Af = 1.1170 5 ^/^ ( 5 : + 5 .)/x'^. (184)
The luminosity formula takes the form
L =
K -f" ^2
(I8S)
350
STUDY OF STELLAR STRUCTURE
It is, of course, clear that a specified 8i(> So) will lead to a unique
value for Sj. Thus, for the complete point-source model with k =
constant, we have an (L, M, x , ijl) relation quite analogous to those
for the other models which have been considered.
An important point to note concerning this complete point-source
model is that, as r — ► o, p o and T—^To (cl. Eqs. [123] and [i 24]).
This clearly shows that the radiative gradient becomes unstable be-
fore the center is reached.
P. C. Keenan has integrated the differential equations I and II
and has obtained by numerical methods several solutions cor-
responding to the model just considered. Table 22 summarizes the
results of Keenan’s computations.
TABLE 22
Solutions for the Complete Point-Source
Model with ic= Constant
Si
Si
M 3
O'*
4 '’"
-li-
st "f" Sjt
16 . g
S-iS
56 . 0
I .ysXios
0.254
29.0
14.4
184
8.00X 10-''
0.331
42.0....
25 -4
3«3
i.8«Xio^*
0.374
54-3
35
600
5. 12X10^*
O.300
7 .L 1 . • ■ .
52.7
1020
S ^>3X10'’
0.410
107. . . .
S4.9
1054
i.iiXio 7
0.442
O25 .0. . . .
596 . 0
31500
1.07X10^
0.475
ii) The point-source model with point muss at the ccnlcr. If we
use the other types of solutions (i.e., solutions which begin as those
of type 3-1 but are types 21, or 31, as / — > «>), then it is clear that
X CO as /'—> o. Further, for solutions of type 21, or 31,, dz/dl ~> o
as / 00. Hence, by the mass relation (89) it follows that
lim M (r) =
This point mass at the center docs not necessarily imply that
p 00 as /• — > o. Indeed, along solutions of type 21,, p o as
r o, though along solutions of type 31,, p — > «» as well. It is, how-
ever, difficult to interpret these solutions without an adequate
examination of the way in which the physical situation alters when
STELLAR MODELS
351
new equations of equilibrium arc introduced as the instability of the
radiative gradient sets in.
4. The point-source model with negligible radiation pressure and
with K = KopT '~''^-^. — The structure of stellar envelopes with negli-
gible radiation pressure has already been investigated in chapter
viii. In particular, the temperature and the density distributions in
the outer parts arc governed by (Eqs. [66], [67] and [53], viii).
and
P
k R \r ’
32 \i7 a i? / \ k s) V KoL ) [r ~
(187)
(188)
For a star of prescribed 7 ^, p, and fc„L, equations (187) and
(r88) give the initial variations of density and temperature. These
equations, however, cease to be good approximations after we have
traversed the outer 10 per cent of the mass of the star. On the other
hand, we can continue these solutions inward, allowing for the varia-
tion of M(:r). Since we are considering a point-source model with
negligible radiation pressure, the equations of equilibrium which
should be used to continue the solutions (1187) and (188) arc:
and
k d , ,,,, GMir)
P 7
(189)
,lM{r)
dr
4 Trrp
(190)
;; i ('/’•') =
p**
47rrT'' ’
(191)
As we have seen, when the variations in M(r) arc neglected, we have
p7"~j -AS = constant . (192)
When the solutions which describe the stellar envelope arc con-
tinued inward by means of (i8g), (190), and (191), the effective
polytropic index, will begin to decrease from its boundary value
352
STUDY OF STELLAR STRUCTURE
3.25. For some definite value of r = r» (say), Wgff will become 1.5.
For r < Tij Weff would be less than 1.5, and according to the discus-
sion in § 3, chapter vi, these regions will be in convective equilib-
rium. The density distribution for r ^ r* should, therefore, be
governed by the Lane-Emden function ^3/2. For prescribed values of
M, Rj and 11 and an arbitrarily assigned value of KqL, we cannot, in
general, fit the outer envelope on to a polytropic core of index n = f .
For, an assigned value of KqL will lead to definite values for p, P,
and M(r) at the interface r = n. Let these quantities have the
values Pi, Pi, and Miu) at r = fi. Now, if the convective core is to
be described by the Lane-Emden function 6^/2, then the quantities
— 503/yg' and —^6^6 should have the following values at the inter-
face:
/|^\ ^ 2 GM{ri)pi
V e Jr^n 5 ^iPi
(194)
(The foregoing equations follow from equations [8], [lo], and [68J
of chapter iv.) In order that a solution be possible, the values of
Ui and Vi thus computed should lie on the P-curve in the (?/, z') plane
(cf. the discussion in § 28, iv). This will not in general be the case.
If the values of M, P, and p are prescribed, we must, therefore, ad-
just the value of k^L until the quantities Ui and Vi computed accord-
ing to the right-hand sides of the equations (193) and (194) lead to
a point on the £-curve in the (u, v) plane. This condition will deter-
mine a (koL, M, R, m) relation of the same general form as the cor-
responding relation for the standard model (cf. § 6, vi).
The situation described in the last paragraph can be considered
in the following alternative way.
Equations (189) and (190) can be reduced to the forms
5 ^ .
2 ^ *
(195)
by the substitutions
p = pe(T , T = 2 \d , r = aj , M. = , (196)
STELLAR MODELS
353
if Pc Tc, a, and Mo satisfy the relations
In the convective core we should have
<T = . (198)
Equation figS) will reduce the equations (195) to the Lane-Emden
equation of index n = If p*. and Tc correspond to the central
values, then the appropriate solution for the convective core is
6 = 62/2- Outside the convective core the temperature gradient will
be governed by equation (191). In terms of the d, and ^ variables,
equation (191) takes the form
where
^ 0 ^ 3 P?
i6t ac aT],'^ '
(199)
(200)
Q is thus a numerical constant.
Suppose we assume that the convective core extends to J
At this point the Lane-I^mden function, will be characterized by
definite values for or, and 0 '. Eciuation (199) will then determine
(). We have
0- I),.,,- <-■)
With this value of () we can numerically integrate the equations
(195) and (199) for $ ^ Now for a solution to be physically
significant cr and 6 should tend to zero simultaneously. For an arbi-
trarily assigned initial value of fi, this will not, in general, be the
case. We can, however, adjust until <t and 6 tend to zero simul-
taneously. This is the method which Cowling has adopted in his
treatment of the equations (195) and (199). The value of which
leads to the physically significant solution is
= 1 . 188 .
(202)
354
STUDY OF STELLAR STRUCTURE
Further, at f f i we have
6 i = 0.7878; 6 'i = — 0.3212; \l/i = 0.4534 . (203)
Equation (201) now gives
0 = 0.1968. (204)
Cowling’s integration shows that the boundary of the configuration
is reached at
f = 7.027 . (205)
At ^ we have
^ = 3 1237 . (206)
Hence,
|- = 0.169; ^ = o-i 4 S
In other words, the convective core occupies i6.() i)cr cent of radius
of the star and incloses 14.5 per cent of the mass.
Equation (197) can now be re-written as
„ _ 2 GM
_ JW
47n^, R> ■
From (208) and (209) we derive that
p = ± nr - f N
? •? -
(211)
Numerically, equations (208), (210), and (211) are found to he
^ nil GM ]
jy GM^ I ( 2 T '’I
P.. = 7.954-^,
P<* = 37.0 P .
STELLAR MODELS
355
Comparing the foregoing equations with the corresponding equa-
tions for the standard model (Eqs. [6i], [57], and [56], vi), we
notice that, though this model is less centrally condensed than a
poly trope of index 3, it is yet characterized by a higher value for
the central temperature.
Equations (200), (204), (208), and (209) lead to the following
mass-luminosity-radius relation :
_ 7.5
If we write (213) in the form (cf. Eq. [39])
I ac (GU\ ^ ^ ,,,
43 s [owd^-s KoJ?.. 3 \ ^ / ^ ’
then we should have
I07.S
4’^
I
0.5
or, introducing the numerical values, it is found that
= 3-30 •
(213)
(214)
(215)
(216)
'Fhe value of 97,. for this model is thus seen to be somewhat larger than
the value 2.5 adopted in chapter vi. It should, however, be re-
marked that if we use (213) to determine the hydrogen contents of
stars, the api^ro])riate guillotine factors will be less than for the
standard model (on account of the higher temperatures and lower
densities in the central regions of the model considered as compared
to the poly trope of index n = 3).
One important characteristic of the model considered in this
section must be noticed, dhe luminosity formula (213) derived
for this model will be valid for any stellar configuration (with
negligible radiation i)ressure) in which the energy-generating regions
do not occupy more than a fraction 0.17 of the radius of the star.
The same is true for the distributions of density and temperature de-
rived for this model. The analysis of this model confirms, therefore,
the generality of the conclusions drawn on the basis of the luminos-
ity formula used in the discussion of chapter vii.
3S6
STUDY OF STELLAR STRUCTURE
BIBLIOGRAPHICAL NOTES
Stellar models have been considered by several writers, and the following
references do not exhaust the list.
1. J. H. Jeans, M.N.^ 85, 196, 394, 1925.
2. H. N. Russell, M,N., 85, 935, 1925.
3. T. G. Cowling, M.N,, 91, 92, 1931.
4. B. Stromgren, Zs. /. Ap.y 2, 345, 1931.
5. L. Biermann, Zs.f. Ap.j 3, 116, 1931.
6. E. A. Milne, Zs. f. Ap., 4, 75, 1932.
7. S. Rosselanb, Zs.f. Ap., 4, 255, 1932.
8. E. A. Milne, Zs. f. Ap., 5, 337, 1932.
9. S. Chandrasekhar, M.N.^ 97, 132, 1936.
10. S. Chandrasekhar, Zs.f. Ap.^ 14, 164, 1937.
11. A. B. Severny, M.N., 97, 699, 1937.
12. J. Tuominen, Annales Academiae Scientiarum Fenniace^ 48, No. 16, 1938.
§ I. — References i and 2. The actual form of the analysis and the deriva-
tion of the luminosity formula (39) is new.
§ 2. — References 9 and 10. The investigations of Severny and Tuominen
appeared after this chapter had been written.
§ 3. — ^The discussion of the point-source model given in the text is in part
based on an unpublished investigation of J. von Neumann. See also reference 3.
§4. — ^The discussion in this section is based on a numerical integration
carried out by T. G. Cowling in ilf.N., 96, 42, 1936; see the appendix to this
paper.
CHAPTER X
THE QUANTUM STATISTICS
In this chapter we shall consider the quantum theory of an ideal
gas, with a view toward the applications contained in the next chap-
ter. It was originally intended to make the presentation of sta-
tistical mechanics as logically satisfactory as that given (following
Caratheodory) of the foundations of thermodynamics in chapter i.
This intention, however, had, in part, to be abandoned, owing to the
space which such an exposition would require; such a discussion
would, also, lead us too far from the main thesis of the present mono-
graph. The most important formula to be established is the relation
between the electron pressure, P, and the electron concentration,
for a completely degenerate electron gas. This formula can be de-
rived in an entirely elementary way, but to appreciate fully the
physical meaning of the (P, n) relation and the physical circum-
stances under which it is applicable a more elaborate treatment is
required, which follows the elementary derivation contained in § i,
below. Applications of the physical theory presented here arc con-
tained in chapter xi.
I. A coviplctely dcgcncrale electron gas: elementary treatment.' A
given number N of electrons can be confined in a given volume V by
one of two methods: either by means of “i)otential walls” such that
electrons inside the “potential hole” cannot escape, or by means of
imposing a certain periodicity condition. We shall consider these
restrictions in greater detail in § 2, but the essential result is that we
can label the possible energy states for an electron inside a given
volume V by means of quantum numbers in somewhat the same
manner as the quantum states of an electron in an atom. If we as-
sume that the volume V is large, then it follows from the general
357
358
STUDY OF STELLAR STRUCTURE
theory that the number of quantum states with momenta between
p and p + dp given by
^ -IT-
The meaning of (i) is that the accessible six-dimensional phase-
space can be divided into “cells” of volume h? and that in each cell
there are two possible states. Now the Pauli principle states that no
two electrons can occupy the same quantum state. This implies that,
if N (p)dp denotes the number of electrons in the assembly with
momenta between p and p + dp, then
N(p)dp < V . (2)
Now a completely degenerate electron gas is one in which all the lowest
quantum states are occupied. In other words, we should have
- 1 ' ¥ ■ (3)
It is clear that if there is only a finite number, N, of electrons in the
specified volume, then all the electrons must have momenta less
than a certain maximum value, /»„, such that
Stt ,
(4)
V fr'
(s)
Ihe threshold value, p„, of p is related to the electron concentra-
tion, n, by
8 t
Zh
:,Pl-
(f>)
'To calculate the pressure, we recall that by definition the pressure,
P, exerted by a gas is simply the mean rate of transfer of momentum
THE QUANTUM STATISTICS
359
across an ideal surface of unit area in the gas. From this definition
it follows quite generally that
PV = I rNip)pv^p , ( 7 )
3 Jo
where Vp is the velocity associated with the momentum p. According
to (3), we have for the case under consideration:
_ Stt
~ 3^’ Jo
p' If <^p >
( 8 )
where E is the kinetic energy of the electron which has a momentum
p. Finally,, if [/kin is the internal energy of the gas which is due to
the translational energy of the motions of the individual electrons,
we have (quite generally)
^ki,
-X
N{p)Edp
(9)
or for the completely degenerate case,
Stt f‘‘«
f/ki,. = I ^^t>‘dp.
From (8) and (10) we find
P =
Ukin
V •
(10)
(ll)
So far the results are quite general, in the sense that we have not
introduced any relation between E and p. According to the special
theory of relativity, we have
(12)
(13)
which gives
36o study of stellar STRUCTURE
Substituting C13) in (8), we have
P =
St
r* ° p^dp
j. (-s"'
Introduce the variable, 0, defined by
sinh d =
me
sinh 00
- A
me
Equation (14) now reduces to
P =
3*3
sinh 4 6 dd .
On integration, we have
p _ STm^c^ |~ sinh 3 6 cosh 6 _ 3 sinh 26 3^
3^' L
16
■
>Je^eo
Finally, writing
we have
where
me
^ = 6.01 X 10” /W ,
f{x) = x{2X* - 3) (a:* + i)'/^ + 3 sinh-' x .
Again from (6), we have
Stjw^c’
n =
= 5 • 87 X io’» .
(14)
(rS)
(16)
(17)
(18)
(19)
(20)
(21)
Equations (igj, (20), and (21) represent parametrically the equa-
tion of state of a completely degenerate electron gas. From (ii) it
now follows that
where
Ukin
trm*c^
3*3
Vs(x:) ,
(22)
«(^)
8 a: 3 [(a:» + i)'/-" - i] - f(x) .
12 ")
THE QUANTUM STATISTICS 361
Equation (22) for the internal energy of an electron gas was first de-
rived by E. C. Stoner. In Table 23 the functions /(*) and g(x) are
tabulated. The table is more accurate and more extensive than any
that has been published so far.
TABLE 23
The Pressure and the Internal Energy of
A Completely Degenerate Gas
X
fM
i(x)
«(*)
/(*)
X
/(»)
«(*)
six)
fix)
0
0.
0.
i-S
2.7..
95.1793s
200.7327
2 . 1090
0. 1 . . .
0.000016
0 . 000024
i-S
2.8..
110.8207
233.7072
2.1269
0.2. . .
0.000505
0.000762
1-509
2.9..
128.3012
275.1070
2.1442
0.3.. .
0.003769
0.005742
1.5233
30..
147.7578
319.2942
2.1609
0.4. . .
0.015527
0.023914
1.5402
3.5..
279.8113
625.728
2 . 2363
0.5.. .
0 . 046093
0.071941
r . 5608
4.0..
484.5644
1114.466
2 . 2900
0.6. . .
0. II 1 126
0. 17604
I . 3841
4.5-.
784.3271
1846.997
2.3543
0.7. . .
0. 23IQQ2
0.27348
1.6099
5.0..
1205.2069
2893.813
2. 4011
0.8. . .
O- 435863
0.71358
1.6372
55 ..
1775.1094
4334 407
2.4418
0.9.. .
0.755661
I . 23849
I . 6654
6 . 0 . .
2525-7390
6237.275
2.4774
1 .0. . .
I . 229907
2 . 0838
1.6943
6.5..
3491.599
87.59.913
2 . 5089
1 .1.. .
I .902586
3.2788
1.7233
7.0..
4710. 192
11948.818
2.5368
1.2...
2.82298
4 . 9468
1.7523
7.5..
6222.021
15939.488
2.5618
I. 3 --
4.04557
7.2052
I . 7810
8.0. .
8070 . 587
20856.421
2.5842
1.4...
3.62991
10.1857
i .8092
8.5..
10302.39
26833.12
2 . 6045
i-S---
7-64033
14.0344
1.8368
9.0..
I . 296694 X 10^
3.401207X10**
2.6230
1.6...
10. 14696
18.9115
1.8638
9.5..
1 .611672X lO^
4.254479X10**
2 . 6398
1.7...
t 3 . 22354 )
24.9920
I . 8900
10.0. .
I . 980725 X 10’
5.2591 Xio*
2.6552
1.8...
1 6 . 94(;69
32.4649
1.9154
20.0. .
3 . 192093 X lO-"!
8.9839 Xio-"*
2.8144
1.9...
21.40937
41.5338
I . 9400
30.0. .
1 .618212X lO*^
4.6494 Xio*^
2.8732
2.0...
26.69159
52.4168
I . 9638
40.0. .
5. 116812X10**
1.48596 Xio’
2 . QO4I
2.1...
32.89010
65.3462
1.9868
50.0. .
r . 249501 X io 7
3 -f> 5 i 5 Xio’
2.9224
2.2.. .
40.10347
80 . 5689
2.0090
Oo.o. .
2 . 591 280 X io7
7 6053 Xio’
2.9349
2.3...
48.43509
98.3463
2 . 0305
70.0. .
4.801018X107
1.41346 Xio®
2.9441
2.4...
57.96311
1 1 8 . 954 1
2.0512
80.0..
8. 190727X10’
2.41703 Xio*
2.9509
2.5...
68 . 89053
142.6823
2.0711
00.0. .
13.12039 Xio’
3.87876 Xio*
2.9563
2.6.. .
81 . 24509
.83.^5
2 . 0904
100.0. .
19.9980 Xio’
5.9206 Xio*
2 . 9606
The function /(x) has the following asymptotic forms:
f(x) ~ gxs — :}x’ + ■Jx» — + ... . (x— > o) (24)
and
f{x) ~ 2;r'' — 3.v“ + . . . .
(.r 00 ) . (25)
362 STUDY OF STELLAR STRUCTURE
Finally, we see that
fix)
2X^
< I
for all finite x .
(26)
The inequality (26) is a strict one. If only the first terms in the
expansions (24) and (25) are retained, we can eliminate x between
(19) and (21) and obtain the following explicit forms of the equations
of state for the two limiting cases:
and
(.T -> O) (27)
(.V— >co). (28)
We may note that g(x) has the following asymptotic forms:
g(x) ~ - {2:7 + .1^^) _ _|. (.V o) (29)
and
g ( x ) ~ 6 x ^ - 8 .V 3 + 72-7 (2: 00). (30)
From (24), (25), (29), and (30) we infer that
Ukin = 2 FV (.r — » o) (31)
and
Ukin = 3 PV (2; — > 00 ) . (32)
The elementary derivation of the equation of state of a comiik'tely
degenerate electron gas should be supplemented in two ways: lirst,
by the enumeration of the states which leads to (i) ; and second, by
the investigation of the physical circumstances under which the
equation of state given by (19) and (21) can be considered to be
valid. These require a rather elaborate treatment of statistical
mechanics, which will now be given. I’or a more general discussion
tha,n that undertaken here, reference may be made to Jordan’s l)ook
which is referred to in the bibliographical note at the end of the
chapter.
THE QUANTUM STATISTICS 363
2. The enumeration of the quantum states . — The wave equation of
the electron in free space is, according to Dirac,
/ W\
I o.xPx "I” ^vPv “h dtumc H ^ j \p = o , (33)
where ay, a„ and a.,,, are anticominuting variables whose squares
are unity, i.c.,
= 28^,. (ju, v = x, y, s, m) , (34)
where 5 ^,. is the Kronecker symbol:
8 ^^ = 0
(m 5 ^ y) , 1
(35)
= I
Further,
•v d
<1
d
(36)
P, = ^ ;
/>..= -eft-
and
w-mi,
(37)
where Pi is the Planck constant divided by 27 r. The wave equation
can therefore be written as
..fid
’'‘{cJl
+ a,„mc j t/' = o .
As is well known, the a’s can be represented as matrices with four
rows and columns and ip is to be regarded as a (complex) vector
with four components, ('housing a particular representation for the
a’s we may write ec[uation (33) as
364
STUDY OF STELLAR STRUCTURE
where rj/a, 1^3, and are the four components of the wave func-
tion. In (39) the matrix representing the a’s should be multiplied
by the matrix |^x| — of just one column — according to the law of
matrix multiplication. Explicitly, (39) takes the form
h
'Pa
-if 4
w
c
h
4" Px
+ Pv
iPz
—if a
h
Pi
if.
^3
'Pi
+ pz
! 'Pl
+ me
yp 2
-'Pi
“^2
-'Pa
= o .
(40)
According to (36) and (37), the foregoing equation is equivalent
to the following four ordinary partial differential equations:
(7 1 + ^ ° ’
(7 1 + ^ ° ’
(7 1 ~ 'f'* ~ {yx + ° •
To solve the foregoing equations, put
= axc® (X = I, 2, 3, 4)
(41)
(42)
(43)
(44)
(45)
where p^, p„, and Pz are now ordinary real numbers and the n^’s are,
for the present, arbitrary numbers. On substituting (45) in equa-
THE QUANTUM STATISTICS
tions (41), (42), (43) j (44) we find that the a^s must satisfy
the following set of homogeneous linear equations:
— + mcj ai
+ pza^ + {pz - ipy)a^ = o ,
+ me) a2+ {px + ipy)
pzd^ O i
pzdi + {px — ipy) a2+ — mc^
{px + ip^di
+ (~ — = o .
In order that the ax’s shall not be identically zero it is necessary
that the determinant formed by the coefficients of the ax’s in (46)
be zero. The determinant is found to reduce to
(f + (f “ • (47)
Hence, the condition that the ax’s do not vanish identically is
(fr-
pi Pi pi
In other words, the relativistic expression which connects the total
energy, £, and the components of the momentum must be valid in
order that (45) may be a solution of Dirac’s equation.
Further, we find from the lirst two equations in (46) that
.. _ + a^ps - ipy) , ,
"■ - E (49)
me H —
e
<hipx + ipy) — il^pz
me H —
e
366
STUDY OF STELLAR STRUCTURE
The foregoing values of ai and aa, in terms of and also satisfy
the last two equations in C46). Hence, of the four only two can
be arbitrarily specified for a given set of values of (p^, py, pg). Hence,
there are two linearly independent solutions for a given set of values
for the components of the momentum which satisfy (48).
We must now obtain some restriction on the possible eigenfunc-
tions due to the presence of the boundary walls. To obtain these
restrictions quite generally, we shall follow Dirac in his approach to
the problem.
According to the general principles of quantum mechanics, there
must be just exactly as many eigenfunctions as should enable one to
represent by a matrix any function of the co-ordinates which has a
physical meaning. Let us suppose, for definiteness, that each elec-
tron is confined between two boundaries at a* = o and x = 4- 'i'hen
only those functions of x which are defined for o < x < Ig have a
physical meaning and must be capable of being expanded in terms
of a complete set of eigenfunctions. It is, of course, obvious that this
will require fewer eigenfunctions than would be required for the
representation of an arbitrary function. A function l\(x), defined
in the range o <x <lx, can always be expanded in a Fourier series
of the form
00
^ (51)
t^=-co
where the are constants and the are integers. It is clear,
then, that if we choose from the eigenfunctions,
(s=)
those for which
px __ 2Trkx
(^x = ± I, ± 2, . . . . ± 00) , (53)
then, Fx(x) times any of the eigenfunctions so selected can be ex-
panded in a series in terms of the selected eigenfunctions. Thus, the
selected eigenfunctions arc suflicient and arc easily seen to be only
just sufficient for the expansion of functions of the form (51).
THE QUANTUM STATISTICS 367
Similarly, if the y and the z co-ordinates are also bounded, so that
o<y<ly\ o< z<h, ( 54 )
then we should have
py 27r ky p:: 2Trk^
Ji "■ ' Tr ~ ^17 ’
( 55 )
where ky and k. are positive or negative integers. The conditions
(53) and (55) can also be written as
Px
kxh
Pv =
kyh
kxh
(56)
where h is now the usual Planck constant.
We have derived (56) from very general considerations. The fol-
lowing special method of imposing the boundary conditions is
illustrative.
We impose the periodicity condition
.V + /i/.* s + k) = y; s) . (57)
From (52) and (57) it immediately follows that the conditions (56)
should be satisfied. We thus see that the state of an electron con-
fined in the volume h ly L can be specil'ied ])y the quantum num-
bers kjr, ky, and k,, and that to the quantum state ky, k.) there
corresponds the following value for the energy, E:
+ wre-*
(58)
From (56) it follows that by making ly, and 1^ sulTiciently large we
can make the discrete eigenvalues of the momenta py, and ps
lie as closely together as we may choose. We can therefore ask as
to the number of quantum states for the electron corresponding to a
specified energy interval, £, £ + LE, where LE is large compared
to the se])aration between the consecutive eigenvalues for E,
Let Z{E)^E be the number of quantum stales in the specified
energy interval, To find Z{E)AEj wo first consider the
368 STUDY OF STELLAR STRUCTURE
total number J (£) of the quantum states for E less than the specified
amount:
J{E)=j^Z{E)dE. (59)
If we remember that for a given set of values of py, and (which
satisfy the relation [48]) there are two linearly independent solutions
of the Dirac equation, it is clear that J{E) is simply twice the num-
ber of points with integral co-ordinates inside the ellipsoid (58). The
equation of the ellipsoid (58) can be re-written in the form
If flx, ay, and a* are large compared to unity, the number of points
with integral co-ordinates inside the ellipsoid (60) is simply the
volume of the ellipsoid, which is
Hence,
4ir
^ dxO'yCiz '
T(V\ - , (E^ ^ A.’/=
(62)
(63)
By (59) it now follows that the number of independent eigenfunc-
tions (which is equal to the number of quantum stales) belonging
to the eigenvalues of E in the range E,E AE is obtained by
differentiating (63) with respect to E:
Z{E)AE = 2
47rF
(64)
where V = If we denote the kinetic energy by E, we have
E = E tne^ .
(6S)
THE QUANTUM STATISTICS
369
Equation (64) can now be written alternatively in the form
^ (f + ”*) (66)
On the other hand, if p denotes the absolute magnitude of the
momentum, defined by
P^ = PI + PI + PI, (67)
then, according to (48) and (65),
— -|- 2 Em = ; pdp = dE . (68)
Equation (66) can therefore be written in the form
Z{p)dp = pHp , (69)
which is the result quoted in § i (Eq. [1]).
We have derived the result (6q) on the assumption that the elec-
trons arc confined in a rectangular box, but it is clear that the result
should be quite generally true independent of the shape of the
vessel. The most general proof of (66) and (69) is due to Peierls,'
to whose derivation reference may be made.
3. The Gibbs canonical ensemble and its properties. — In the last
section we saw that the number of quantum states with energy
between E and E dEh given by
Z{E)dE = 2 ^ {2mE -f ' dE . (70)
The foregoing density of the quantum states in the scale of the
kinetic energy is, in fact, a very general characteristic for a gas of
material “particles.”^ Let the discrete eigenvalues of the energy E
be denoted by e,, €j, . . . . , «„
> R. Pcicrls, M.N., 96, 7S0, ro3(>. The same result is also olitainecl by K. K. Rroch
{Phys. Rev., 51, 586, ig^;), who has explicitly solved the Dirac equation in a spherical
potential hole and enumerated the states.
* We shall use the word “particle*’ to denote an electron, molecule, or atom. The
theory presented in this section deals with a ^tmeral assembly of similar particles.
370
STUDY OF STELLAR STRUCTURE
Let us consider an assembly of N similar particles in a given
volume V and with an internal energy U due to the kinetic energies
of the individual particles. Now, since the particles are assumed to
be similar, they cannot be distinguished from one another, and a
microscopic state of the gas will be completely described by the
specification of the number of particles, n„ beloirging to the eigen-
value e, for the energy E. We should then have
N = ^ n . (71)
S
and
U — . (72)
A possible sequence of numbers «„ Mj, , must satisfy
the restrictions (71) and (72). We shall write the dilTerent sequences
of values for the m.’s which satisfy (71) and (72) in the form
'^0 1
Til , . . . .
«(l)
• • • )
ffii) «(2)
.
• • • J
1 •
. . . J
. . . ,
(73)
.,(W0
/J(, J ?ll , . , . ,
, nr, .
. . . ,
where W is the number of different solutions in integers for the
equations (71) and (72).
The entropy, S, is now defined by
5 = k log W , (74)
where k is the Boltzmann constant. Instead of (74), we can write
c-'V* = w . (75)
The actual justification of (74) and (75) will take us too far into
the foundations of statistical mechanics in its relation to thermo-
dynamics and for this reason we shall simply assume the validity of
(74) and (75) reference may be made, however, to the literature
quoted in the bibliographical note.
Now the restrictions (71) and (72) can be droi^ped by the passage
THE QUANTUM STATISTICS
371
from the microcanonical state specified by (71) and (72) (accord-
ing to which both N and U are defined exactly), to a canonical
state in which both the energy U and the number of particles N
are distributed canonically, i.e., in such a way that U and N have
sharp maxima at certain prescribed values- say U and A. This
process, due to Gibbs, will become clear from the following discus-
sion where the method of actually carrying out this passage to the
canonical distribution is described. Our presentation closely follows
a treatment originally due to l^auli.
First, let us try to replace condition (72) by one whereby U must
have a sharp maximum at a certain specified value oiU ^ U (say),
while retaining the condition (71). This means that we make
the passage from the microcanonical state in which both N and
U are exactly specified to one in which N has the specified value
(exactly), while U has an extremely sharp maximum at U in such a
way that, as we shall see presently, U is appreciably dilTerent from
zero for
U = U ± UJ ,
where
MJ I
jj ^ vn ■
(7fi)
(77)
According to (72) and (75), we can write, for the microcanonical
state,
.V
iUI
(78)
where d is, for the present, an arbitrary constant. It is clear that in
the summation occurring in the exjionent in the rif'ht-haiul side of
(78) we can choose i to be any number from f, 2, . . . . , ly (cf. the
scheme [73]); and, since for each value of i and /
{i,j = r, 2 IF) , (79)
we can write (78) more “symmetrically" as
s
nil
(80)
372 STUDY OF STELLAR STRUCTURE
We shall now drop the restriction (72) and write, instead of (80),
Q
where the index “q” means that we should now have
= iV for all q ,
(82)
without, however, the restriction (72). Further, the summation with
respect to j in (81) is to be carried over all the different solutions in
integers of equation (82). The quantity ^ is now so chosen, that
the expression (81), when differentiated with respect to U for fixed
^ and fixed 6,’s (i.e., for a fixed V), vanishes. Hence,
I
k
(83)
Now according to the first and the second laws of thermodynamics.
dQ = TdS = dU + PdV ,
so that
Hence, according to (83) and (85),
(84)
(85)
( 86 )
Since the free energy, F, is defined by (cf . § 1 1 , i)
F = U -TS , (87)
we can now write (81) in the form
kT *rZy “ •
=
e
4
( 88 )
THE QUANTUM STATISTICS
373
We shall now show that according to (83) and (88), U has, in fact,
an extremely sharp maximum at a certain U = U (say) and that U
is appreciably different from zero only in a range of AU such that
A17/Z7 ~ i/ViV. Further, we shall show that the entropy, de-
fined according to (83) and (88), differs from that defined by (75)
only by a quantity of the order of log N/N. To prove these, we
first remark that (81) is now interpreted by the statement that the
probability of a microscopic state defined by a sequence of numbers
(«?’> • . • . , . . . .) and an energy U = is proportional
S
to
(89)
In order to obtain the probability of a microscopic state with a
definite total energy U corresponding to the canonical distribution
(89), we have to sum over all sequences which lead to the
energy U.
Let SiiU, V) be the entropy defined according to (75), and
SiiiU, V) that defined according to (83) and (88). By our defini-
tion of Si{U, K), according to (89), the rule stated above for de-
termining the probability of a microscopic state with a definite
energy U, and (80) we find that
•'‘7
W{U) = constant c* . (90)
If we now regard the right-hand side of (90) as a function of U, we
sec that W{JJ) has a maximum at i7 = (say), where
= (91)
We therefore expand the exponent occurring in (90) in the neigh-
borhood oi U = V hy & I'aylor scries in LU = U — U and retain
terms up to the second order in AU:
S,(U, V)
k
- rJU +
(92)
374
STUDY OF STELLAR STRUCTURE
By (90), (gi), and (92) we now have
Pf (AU) = constant e ** . (93)
In writing (93) wc have used the circumstance that is
negative, for, according to (85),
= _JL (^I\
)U^)v T^\du)v'
Hence, if we denote by {^UY the “mean square error,” wc have
IKOf =
)v, v=c
\dU‘jv,u^U
The right-hand side is easily seen to be of order N'"^\ and hence,
VnJ '
In order to prove our second statement concerning the entropy, wc
write (8t) in the form
To carry out the summation in (q 8) we first lix a certain value for U
and select from the sequences those which correspond to a
specified JJ . We then sum over all the possible s. Equation (gtS)
can then be written as
c = V W{U)e~''»^ ,
THE QUANTUM STATISTICS
375
where W{V) has the same meaning as in equations (73) and (75).
Hence,
SlI „ Sj(U)
-T —r eU f V
e* = * . (100)
u
Expanding the exponent occurring on the right-hand side of (100)
in a Taylor series about U = U, yre have, according to (91) and
(92),
^11
^ .rr.i l^\
k _ e ^ v,u=U
a^sj.
(AC/)*
AU
or
Sfj-Sj
^L\r^\
= 2A- I Wv I/, =
AV
(AC/)*
(lOl)
(102)
Replacing the sum by an integral, wc have
c k
^ ^ ^ 2^1 Gc;*) V, c/=c 7 1 (/(AC/) , (103)
UJ-°o
or, finally,
f, k =
N
C 7 .
2 irk
/^A
\c)C/'' Jv,u
(104)
Equation (104) is equivalent to
Sii — 5 /
Si
Further the maximum error in determining tJ according to (83)
(instead of according to [91] with Sj instead of S/r in [83]), will also
be of the order (105).
Second y we now try to replace the condition (71) or (82) by the
one that N is to have a sharp maximum at a certain specified
376
STUDY OF STELLAR STRUCTURE
N = N (say), in such a way that, as we shall see, N is appreciably
different from zero for
N = N ± ^N ,
where
AN I
N Vn '
(io6)
(107)
This corresponds to the passage to a canonical distribution, not only
for the energy U but also for the number N of the particles con-
cerned.’ To do this we proceed as follows:
According to (82) and (88), we have
- 2 -
(<7)
(108)
where a is, for the present, an arbitrary constant. 'Jb make the
passage to a canonical distribution, we write, instead of (io8).
r2:('
(109)
where we no longer have the restriction (82) but a is now so chosen
that the expression (109), when differentiated with respect to N for
fixed temperature T and volume V vanishes. Hence,
We shall now show (following Pauli) that N defined according to
(109) and ^10) has, in fact, an extremely sharp maximum at a cer-
tain N = N (say), and that N is appreciably different from zero only
in the range AN/N ^ i/^N. Again, we shall show that the free
energy defined according to (109) and (no) differs from that defined
by (88) only by a quantity of the order log N/N.
i This is the essential difference between Gibbs’s classical treatment and the quan-
tum mechanical version of Gibbs due to Pauli.
THE QUANTUM STATISTICS
377
To prove these statements we remark that (109) is now inter-
preted by the statement that the probability of a microscopic state
defined by a definite sequence of numbers (Wi, . . . . , . . . .) is
proportional to
e
(Ill)
In order, then, to obtain the probability of a definite total number
AT” = ^ Tis of particles corresponding to the canonical distribution
(in), we should sum over all such ^a-scquences which belong to the
number N.
Let Fi{N ^ r, V) be the free energy defined according to (88) and
FjiiNj T, V) that defined according to (109) and (no). By the
definition of Fi{N, T, V), and according to (in) and the rule stated
above, the probability W{N) for a definite total number iV', is seen
to be
IK(iV) = constants * ^ J . (112)
For a fixed a, W {N) has a maximum where
(113)
Let (113) be satisfied at A/' = (say). We now expand the expo-
nent occurring in (112) by a Taylor series in the neighborhood
N N \\\ terms of AA/” = N — N and obtain
a + i (1^') ClN + -^.{ ) JANy
kT\dN/N=N_ 2 / 5 ! Jn---n
+
(114)
By (112), (113), and (114) we now have
IK(AA) = constant c
_ (A/V)=
(I IS)
378 STUDY OF STELLAR STRUCTURE
Hence, the “mean square error,” (AiV)*, is given by
which proves (107).
In order to prove the second statement concerning the free energy,
we write (109) in the form
= 4-1 • *r;
(118)
To carry out the summation in (ii8), we first fix a certain value for
N and select from all the sequences those which correspond to a
specified N, We then sum over all the possible N's. Equation (i i8)
can then be written as (cf . Eq. [88])
77
« L
kT
- 2 -
Fli/v, r, n I
Tf J
(119)
Expanding the exponent occurring on the right-hand side of (iiq)
as a Taylor series about N = N, wc have, according to (113) and
(114),
_ FjiN,T,V) I
g kT = V
f
AN
or
Pjl-Fj j /f^^Fj\
AN
JlR^ V
^ kT
-2<
(120)
Replacing the sum by an integral, we have
THE QUANTUM STATISTICS
379
or
Fji-Fi
e =
Equation (123) is equivalent to
Fit — Fi
2irkT
(123)
Ff
■-f- Tisf
^[dJV^/r,r.Jv=Jv
Further, the maximum error in determining a according to (no) in-
stead of (113) with Fi (instead of F/r in [no]) will also be of the
order (124).
We shall now return to (109). Since there is now no restriction
with regard either to or to ^n.e., we can re-write (109) in the
form
(125)
Now equation (no) is to serve the purpose of determining a. We
can transform this into a more convenient form as follows: Dif-
ferentiate (125) logarithmically with respect to A keeping V and T
constant. Then,
I
(^)r. . + ^
■yjlf
. (126)
Hence, according to (no), we have, since (Oa/dN) is not in general
zero.
380
STUDY OF STELLAR STRUCTURE
The thermodynamical significance of a can be found as follows:
As we shall see presently, {F/N) depends, apart from temperature,
only on the concentration of the particles N/V. Consequently, we
can write
II
M
to
Hence,
\dNlT.v ^ \V' V a(,N/V)'
(129)
But by (128)
(dF\ m dj
KavjN.T v^diN/v)'
(130)
From (129) and (130) we derive
KdN/T.v \dV)n,T
(131)
Since, however, we have the thermodynamical relation (chap, i,
Eq. no).
we have
(133)
where G is the thermodynamic potential at constant pressure (cf.
§ 12, i). Hence, by (no) and (133),
^ G_ _ F + FV
“ NkT NkT ’
which then gives the thermodynamical meaning of the parameter a.
For the calculation of the statistical mean value of any physical
quantity it is important to note that the quantity
(■ 35 )
THE QUANTUM STATISTICS
381
which occurs in (125), is, apart from a constant, the probability of a
definite microscopic state:
W{nj, «2, = constant . (136)
If we compare two microscopic states («„ and
{n'x, . . . . , n's, . . . .) for which the total number of particles (71)
and the total energy (72) are equal (or nearly equal), then, accord-
ing to (136) , the two states are equally probable; this is, in fact, an as-
sumption implicit in equation (74).
For the internal energy U we have, immediately.
(137)
Equation (137) follows also from the thermodynamical relation
(chap, i, Eq. [no])
U =
(138)
On the other hand, differentiating (i 25) logarithmically with respect
to T and keeping N fixed, we have
By (127) the terms proportional to (Oa/dT) cancel, and (138) and
(139) together imply precisely the expression (137) for U.
382 STUDY OF STELLAR STRUCTURE
Finally we shall obtain some formulae which
portance in the application of the theory.
By (134)
are of practical im-
(140)
Hence, according to (125), we can write
PV = kT'^ log <r. ,
(141)
where
O-g = .
(142)
Equations (127) and (137) can now be written in the form
^ ~ 2 £ (H3)
and
Equations (141), (143)1 ^■ncl (144) arc extremely general and give
the physical variables for a system in statistical equilibrium which
is also a thermodynamical system.
4. The symmetrical and the antisymmelrical stales; the Einslcin-
Bose and the Fermi-Dirac distributions.- If we consider a system con-
taining a number of similar particles, then no observable change is
made when two of them are interchanged. A satisfactory theory,
then, should consider two such observation ally indistinguishable
states as really the same state.
Suppose we have a system of A similar particles. Let?„v„ . . . . ,
gv, be the variables describing the first, the second, the Ath
particle in the system. Then the Hamiltonian, H, of the system
will be a function of the variables ^2, . . . . , g^;
H = g,; . . . . ; . (i 4 S)
THE QUANTUM STATISTICS
383
Since the particles are indistinguishable from one another it is Hea r
that H should be symmetrical in all the particles, i.e., symmetrical
in the variables qi, , q/f. If is a wave function describing the
system, then we should have
= 0. (146)
From the foregoing it follows that if , gv) is a solution of
(146), then so is q^, .... , qn), where stands for the
function obtained by applying the permutation P to the variables
?> 9jv-
Suppose that at any given time,
= (147)
then, since H is an operation in the space variables only, we have
HFif = [H]o = o , (148)
so that by (146) d(P^)/dt s o at / = If, now, H and P^ are
analytic functions of t for all real values of I, it follows that we can
prove by repeated applications of the argument that
B '‘ ^ ’ (^49)
for all n, and that therefore P'l' = o for all time. From this it follows
that if \F is of a given “symmetry character" at a given instant of
time, it retains its “symmetry character" for all time. In partic-
ular, if the wave function is initially symmetrical (i.e., is unal-
tered by any permutation of the variables g„ . . . . , q^,), then it
is symmetrical for all time. In the same way, if the wave function
is initially antisymmetrical (i.e., is unaltered or changes sign accord-
ing as an even or an odd permutation"* is applied to the variables),
then it is antisymmetrical for all time.
4 A simple interchange is an odd permutation, while two interchanges will be an
even permutation.
3^4
STUDY OF STELLAR STRUCTURE
The permanency of the symmetry properties of the state means
that for some kind of particles only the symmetrical or the anti-
symmetrical states occur. It is found that light quanta should be
described by symmetrical wave functions (as we shall see, it is only
then that we have Planck’s law for radiation). On the other hand,
the electrons should be described by antis3anmetrical wave functions,
only then can we obtain Pauli’s exclusion principle, which states
that no two electrons can be described by the same set of quantum
numbers. For if two electrons were described by the same set of
quantum numbers, then an interchange of the variables correspond-
ing to these two electrons must leave the wave function unaltered ;
the wave function under these circumstances can vanish identically
only if it is antisymmetrical in the variables of the two electrons.
Since the “two electrons” can be any two, the wave function must
be antisymmetrical in all the variables describing the different
electrons.
For our purposes it is only necessary to remark that in the sym-
metrical case there can be o, i, 2, . . . . , », particles in the same
quantum state, while in the antisymmetrical case there can only be
o or I particle in a specified quantum state. The former case leads to
the Einstein-Bose statistics while the latter case leads to the Fermi-
Dirac statistics.® Hence, according to equation (142) of the last sec-
tion, we have for these two cases,
00
= ——1^— (symmetrical case) (150)
and
<7h = I + (antisymmetrical case) . (151)
s It is somewhat misleading to use the word “statistics” in “JOinstein-Bose statistics”
and “Fermi-Dirac statistics.” There is only one statistics, namely, the Gibbs statistics
described in § 3. The symmetrical and the antisymmetrical cases simply corrosi)on(l to
two different assumptions for the evaluation of cr^ (Fq. [142I); the explicit forms for
N, U, and PV naturally differ, but nevertheless we have the same statistical theory
(Gibbs) underlying both the cases. It would be more logical to refer to “Finstein-Bose
formulae” and “Fermi-Dirac formulae.”
THE QUANTUM STATISTICS
38s
From (150) and (151) we have, respectively,
and
I
ga+iJe^ qp j
(152)
d , _ e„
dd + I
(153)
In (152) and (153) the minus sign corresponds to the symmetrical
(Einstein-Bosc) case and the plus sign to the antisymmetrical
(Fermi-Dirac) case. Finally, according to (141), (143), and (144),
we have
If- = + log (i + «-(“+■»«.)) , (154)
and
I
q: I »
t/ = V.
I « X
« -[- I
(iss)
(156)
5. The electron gas: general formulae. h’or an electron assembly
we should use the results for the antisymmetrical case considered in
§4. The summation over “s" occurring in equations (154), (155),
and (156) can be transformed into integrals if we remember that the
density of the quantum states is given by Z{E) (the explicit expres-
sion for which is derived in § 2) :
N =
T” ZiE)dR
Jo -H I =
T” Z{E)EdE
e-+TH q; j .
(157)
e-(-+^i<)]Z(E)dE .
(158)
STUDY OF STELLAR STRUCTURE
■ 386
The expressions take their simplest forms when, instead of the
kinetic energy E we choose the momentum p as the variable for
the integration. Then, according to (69) and (68),
Z{p)dp = pHp , (159)
where
— + 2Em = p^ . (160)
Equations (157) and (158) can now be written in the forms
§2^ r°° p^dp . n-^ r_^^dp
jo 4- I ’ A3 Jo + i ’
Q— TT’ /'oo
PV = log [i + .
(161)
(162)
Equation (162) can be transformed by an integration by parts so that
PV =
sttf r
p3
3 A 3 Jo + I dp
(163)
The equations for U and P can be derived in an elementary way on
the basis of the distribution function,
Nip)dp =
8tF p‘dp
hi 6“+'’"- + I
(164)
which gives the number of particles in the assembly which have
momenta between p and p + dp. In particular, equation (163) is
consistent with our definition of pressure used in § i , above.
We can obtain (164), or more generally, an e.xpression for the
number of electrons in the assembly with the components of the
momentum in the range {p^, py, p,; p, + dp„ />„ + dpy, p, + dp,),
as follows:
From equations (56) it follows that the number of quantum slates
in the specified range is given by
^ ^ ^ ^P^^Py^P^ • (T 65 )
2
THE QUANTUM STATISTICS
387
The factor 2 in (165) arises from the circumstance that for a given
set of values for px, py, and px the Dirac equation has two (or no)
linearly independent solutions according as U8) is satisfied (or not).
The number of electrons in the range specified is obtained by sum-
ming (15s)) iiot over all the quantum states but only over those in
the range specified. We thus have
dN
V_ dpxdpydpx
_|_ j >
(166)
which expresses the Fermi distribution for the momentum com-
ponents. If a is very large, then we can neglect the term unity oc-
curring in the denominator in (166) and obtain
dN 2 c ® '^^^dpxdpydPx , ( 167 )
which expresses Maxwell’s law of the distribution of momenta. The
case a :» 1 is called the nondcgcncrate case. On the other hand, if a
is large and negative the Fermi distribution becomes markedly
different from the Maxwell distribution and the gas is then said to
be degenerate. We shall consider these questions in greater detail
in the following sections, but we shall now obtain a very convenient
form of the equations (161) and ( i6,0. The transformations to be
introduced are due to Juttner. Let
— = sinh 6 ,
me
(i68)
E
= mc^ (cosh 0 — i) .
(169)
Then we easily derive
oc
II
P sinh=* 6 cosh 6 dO
1 — nisli 0 1
f A ^ ^ ^
O
(170)
U =
s inh^ 6 cosh 6 (cosh 0 — i)d 9
L ctish 0 _l_ y
A ^
(171)
388
and
STUDY OF STELLAR STRUCTURE
P =
Sirm^c^
□o
sinh 4 B dd
JL cosh fl_j_ j
where we have used
JL — aa—dmc^
A~ ^
(172)
(173)
6. The degenerate case . — As we have already pointed out, the de-
generate case corresponds to the case where a is large and negative.
A condition equivalent to this is that A (as defined in [173]) is very
large compared to unity.
It is clear that as A — > 00 the term
I, cosh d
a""
(174)
occurring in the denominator in (170), (17 1), and (172) is negligible
compared to unity for all 6 ^ 60 where do is defined by
log A = cosh do . (175)
We can therefore write as a first approximation (a rigorous justi-
fication is given later in this section)
N =
J
roo
sinh^ 6 cosh 6 dd ,
'0
(176)
U =
J
roo
sinh^ 6 cosh d (cosh d — i)dd ,
'0
(177)
P =
Jo
'0
sinh"* d dd .
(178)
The foregoing expressions arc precisely those considered in § i (eqs.
[6], [10], and especially [16]). In order, however, to consider more
explicitly the circumstances under which the foregoing approxima-
tion becomes valid, we shall have to evaluate the integrals (170),
THE QUANTUM STATISTICS
3S9
( 1 7 1 ) , and (i 7 2) to a higher degree of approximation than above. To
do this we shall first prove the following lemma (due to Sommerfeld).
Sommerfeld’s lemma. — If (p{ti) is a sufficiently regular function
which vanishes for u = o, then we have the asymptotic formula
^ + ^[c.-p"{Uo) + (179)
lifhere Uo = log A and c^, q, . . . . , are numerical coefficients defined by
Cv
(180)
The asymptotic formula (179) is valid if wc neglect quantities of
the order 6“"" = A”^
Proof: Split the range of integration at = log A. Wc then have
or
In the lirst integral occurring on the right-hand side of (182) put
u = Wo(i — /) , (183)
390
STUDY OF STELLAR STRUCTURE
and in the second integral occurring on the right-hand side put
U = Wo(l + t) . (184)
We now have (remembering that «o = log A)
du
t \ P — /)]
«5(mo) - Mo ^ ^ di
+ Q]
I +
dt .
(185)
In the first integral occurring on the right-hand side of (185) we
can extend the range of integration to 00 ; this will introduce an
error of the order e”"®, which is beyond the range of accuracy of the
asymptotic formula we are establishing. Hence, wc have
du
d(p{u)
du
— + Uo
+ Q] — <^'[Wo(l — /)]
= V?(«o) + 2 ^
J'=2, 4, 6,
I +
(y
dl
^ r°°
i)! Jo r + c"»'
f// . (186)
On the other hand, wc have
Since the constants are defined according to (180), wc have
~ + 2[Co(p"(Mo) + + ] , (188)
- e- + I
THE QUANTUM STATISTICS
391
which proves the lemma. We may note that
“ 77 » ~ rrr > . (189)
12 720 30,240 ^
In order to apply (188) to an asymptotic evaluation of the inte-
grals (170), (171), and (172), we must first transform them into
suitable forms. Let
cosh 6 = u . ( too')
First consider the integral for P. Then
Sirm^c^ 1
3/^3 ^mc
where we have put
Flence,
<p(uo) = sinh’ 6 du = dmc^ ^ sinh^ 6 dd , (193)
which is an integral we have already computed (Eq. [16]). Intro-
ducing, as before, the variable x which is defined by (Eq. [18]),
"I
du d(p(u)
I « I du
-e“ + i
sinh^ 6 .
a; = — = sinh ft, ,
me '
!?(«..) = ^^f{x) ,
where /(,v) is defined as in equation (20). From (192) we derive that
(S)„=„„ = (^96)
( 3 cosh ft, . . , ^ V
V/w7„=,,„ {{fme^y sinh^ ft,
— 3 (a-^ + ~~ i)
392
STUDY OF STELLAR STRUCTURE
By Sommerfeld’s lemma it now follows that
P =
3/t3
; fU\ Tt 4 - x{x^ +
(d-mc^y f{x)
+
7ir^ {x^ + iy^^(2X^ - i)
x^fix)
+
....].
(198)
To evaluate the integral for N we write it as
N =
Stt V
where (p{v) is now defined by
f oo
d
I
r-
dii d(p{u)
, du
+ I
(199)
d<p{.u) _ j
du
= \ sinh 2d .
From (200) wc derive that
and
Wc thus have
SttFwV
N = n —
f V dmc^ ,
^(«o) = ,
/ _ I 2.r^ + I
\du^)u^u,, X
( 3 £
\duyn=uo {dmc^y x^ ’
3^3
I +
2X^ + I
77r^
{dmc^y^ 2X^ ' /^o{dmc^Y
(200)
(2or)
(202)
(203)
(204)
The integral (171) for U can be evaluated similarly. We liml that
TrVm^c^ . . '
4^r* (32:^ + i)(x^ + i)*^^ - (2.r^ + i) ,
{■dme^y x/;(x)
. (20s)
where g(x) is defined as in equation (23).
On comparing (198), (204), and (205) with the corresponrling
expressions (19)) (21), and (22), we see that the dominant terms in
THE QUANTUM STATISTICS
393
the present expansions agree with our earlier expressions for com-
plete degeneracy. On the other hand, we now see that the necessary
condition for the convergence of the foregoing expansions is that
-f-
f{x) ^
(206)
and
+ I „
{Sme-y a.v-* ^
(207)
(.3.V" -1- -f — (2.r^ -f- 1 )
{i3^mc‘y xg{x) ^
(208)
As a: — » o, the foregoing inequalities take the limiting forms (cf.
Eqs. [24] and [29])
(dme^y 2X'<
I
{dme^y 2X'*
«i ;
-^«i
2X^
(209)
Again, as the inequalities (206), (207), and (208), take the
limiting forms
TT- £ . TT- _i_ _jr‘ 2 . .
From (209) and (210) it is clear that a necessary and sufficient condi-
tion for the sclting-in of degeneracy is
47r" :\ix‘ + i)’/"
(ifme-y fCv)
« I .
(211)
The inequality (an) imf>lies the other two ([207] and [208]).
In using (211) as a criterion for degeneracy, it should be remem-
bered that X is related (in a first approximation) to the mean con-
centration, n, of the electrons by
n
(212)
Further, ^ is r/kT.
For a,stronomical applications, the criterion of degeneracy is
394
STUDY OF STELLAR STRUCTURE
stated more conveniently in a somewhat different form, which will
be obtained in the next chapter.
We shall conclude the discussion of the case of degeneracy with a
derivation of the specific heat, Cy, of the electrons at constant
volume. To evaluate Cy we note that, by definition,
Cy =
(213)
According to (204), the condition of the constancy of N is equiva-
lent to
XdTjv * (.214}
By (20s)
/ dU\ _ TVm^c^
XdT/v 3^3 \^dx xdTjv
(3;^^ + + 1)^^^ — (2.v^ + i) "
T{^mc^y X
It is easily found that
^ - i] . (216)
By (214), (215), and (216) we have
Cv =
Stt^ V
- xix^ + i)*''" ;
or by (204), the specific heat per electron is given by
Cy __ TT^k^ (X^ -f- y,
N ?nc^ .V*
(217)
(2t8)
7. The nondegenerate Maxivell-Jutlncr case. Let us now consider
the other limiting case when A"* is very large c<)mi)are(l to unity.
We can then neglect the unity occurring in the deiKuninators o[ the
integrands of (170), (171), and (172). d'he present case is therefore
THE QUANTUM STATISTICS
39S
the opposite extreme to the one considered in § i. The integrals for
N, P, and U can now be written as
,, 2>TVin^c^ , i
*■
/^CD
cosh a 0 cosh 0 d0 ,
’o
(219)
^00
I g-amflemha 0 0 (cosh 0 — i)d 0 ,
(220)
and
PV = A I
3A’ J
1 sinh4 e Je .
O
(221)
The last integral can be simplified by an integration by parts. We
find that
Stt V I ^00
PV = — - A 1 sinh' 0 cosh 0 d 0 .
(222)
Comparing (219) and (222), we sec that
N = PV& or rV = NkT . (223)
In other words, Boyle's law is idcnlically true for the nondegenerate
case. This important result was first established by Juttner, though
it is implicit in some earlier work by Planck.
The integrals occurring in (219) and (220) can be evaluated
explicitly in terms of Bessel functions. We use the formula®
cosh vO (Id
Kfz) ,
(224)
where A,(s) is related to the Hankel function with imaginary argu-
ment as
Kfz) - . (225)
Since
sinh" e cosh 9 = :l(cosh ^9 — cosh 9) (226)
and
sinh^ 9 cosh^ 9 = J(cosh 40 — i) , (227)
^ 'I'hc formuliic arc contained in G. N. Watson’s Hexscl Functions, Cambridge; see
pp. 79, 181, and 202 in Watson’s book. Kquation (224) is due to Schliiili.
396 STUDY OF STELLAR STRUCTURE
we have, according to (219), (220), (224), (226), and (227),
N = A i[K,{dmc^) - ,
^ ~ — K„(dmc‘‘)] — WK^idmc^) — A,(«>otc»)]1 . (229)
Using the recurrence formula,
we find that
- K^^{z) = -~ K,{z ) ,
K,{z) - K,(z) = 2 K,(z)
Uz) - Ko{z) = ; bATjCz) + A:.(2)] . (232)
Z
Equations (228) and (229) can therefore be simplified to the forms
N A — (233)
^ A [J {3A:j(tymc’) 4 - Kii^mc ^) } - A%(«>7«c^)] . (234)
From (233) and (234) we find that
= N ^
By (223), equation (235) can also be written as
-f- Ai(??7WC^)
-- I . (236)
If dme^ )$> I wc can use the asymptotic formula
f.\ ( -1_ , (4>'“ — iQ Uv’ - 3*)
■+^+
21(82)^
+ (237)
THE QUANTUM STATISTICS
397
Equation (236) now reduces to
^ ~ • ("38)
The inequality i}mc^ » i is equivalent to the condition
kT « mc‘ or T « 5.90 X 10® degrees Kelvin. (239)
On the other hand, if dmc^ « 1 (i.e., T » 6 X 10®), we should use
KXz)
1 (»' ~
2 (Iz)'
From (236) and (240) we find that
(240)
_U
PV~^^ ’
» ,
(241)
Wc thus sec that U/PV varies from i .5 to 3 just as in the completely
degenerate case. Here this variation is associated with increasing
temperature, while there it arose because of increasing density. In
either case the change of the ratio (U:PV) from 1.5 to 3 is associated
with an increasing number of electrons in the assembly with veloci-
ties approaching that of light. In 'Fable 24, due to Chandrasekhar,
the ratio {U :PV) as a function of dnic^ is shown.
TAUI.K 24
The Internal Knkrgy and the Si’h( ii.tc Heat oe a I'ekeect Gas
From (235) we now derive that
i. (—\ — ^ \?,Ki {dtn c^) -j- K,{dmc^y ^
N \dT )r, V 4 [ Kl(dmc^)~ J ~ dT ’
398 STUDY OF STELLAR STRUCTURE
or, writing z = we find
I 4.[3^3(zy;4i(£)
i^dT Jt, V
sL 4^2(2)
]
(243)
Thus the specific heat (per electron) at constant volume is given by
\ 3K, + K, , _ 3E^±E1^
Nk aV Ki J ■
Using the formula
= -2^;,
we can re-write (244) as
(244)
(24s)
Cv _z^\2iK, + K,) + (K, + Ko) (3K, + KdiK^ + K,)^ , ,,
m-&[ T. Ti J- ("46)
By using (237) and (240), we can show that the quantity on the
right-hand side of the foregoing equation varies from 1.5 to 3 as
^mc‘ decreases from infinity to zero. More directly, from (223),
(238), and (243) we derive that
or
(246')
(247)
In Table 24 the quantity Cv/Nk is tabulated with xhne:^ as argu-
ment. Since Cp— CV = Nk, it is clear that the ratio of the specific
heats varies from 5/3 for T « mc^/k to 4/3 as T » mc^fk.
8. The nondegenerate case: a second ap[)roximalion. If the ex-
ponential terms occurring in the denominators of the integrands in
(170)1 (171). and (172) are large (but not infinitely large) compared
to unity, we can expand
THE QUANTUM STATISTICS
399
as an infinite series and obtain
^TrVmH^
N =
U =
and
PV =
STrVm^c^
V(-)»+'A» 1
^00
I ^-ndtne^ cosh 9 ginh^ 0 COS B dS ,
(248)
00
1
^ (-)»+■ A” J
1 g-»,yw^2cosho gjnh^ 19
*o
X cosh 6 (cosh B — i)dB ,
■ (249)
^(-)“+'A’‘J
^co
I g-namc^ cosh B sij,h4 $ dd .
*o
(250)
Equation (250) can be transformed into
PV =
sttFw^ I
^ -iW n Jo
hi § n
W =1
cosh 0 ginh® B cosh B dQ . (251)
I'hc integrals occurring in the foregoing equations arc of the same
form encountered in § 7. By (224), (226), and (227) we now have
A =
Stt Kw’e’
Sc-)"'"'
«== I
ndme^
(252)
PV =
StrVin^c^ I V"'
hi d
4 V (-)'•+■ 4”- K,{ndmci) ,
> n^dme^ ^ ''
(253)
and
V = ^ - 1) «+i +K,{ndmc‘)\
— Kz{n^mc^) 1
(254)
9. The unrclativistic case. So far we have distinguished between
degeneracy and nondegeneracy, but we have allowed in either case
for the relativistic mass variation with velocity. However, in certain
astronomical applications (as in most terrestrial applications of the
STUDY OF STELLAR STRUCTURE
Fermi-gas laws) it is permissible to neglect the relativistic effects
and write with sufficient accuracy
^ = = Pdi’ ■ ( 255 )
Inserting (255) in our general formulae (Eqs. [161] and [163]), we
obtain
if. (.56)
go+ij/i: ^ j >
pv.ivz r
3/1* ' J. e-*«‘ + I •
Comparing (257) and (258), wc find that
U = %PV (259)
is valid independent of degeneracy conditions provided, of course,
that the relativistic effects are neglected. This is a generalization
of the result we have proved directly for the case of complete de-
generacy (Eq. [31]) and for complete nondegeneracy (Eq. [238]). Put
■&E = u , a = —log A , (260)
and introduce the integral i 7 „, defined by
equations (256), (257), and (258) can now be written more con-
veniently in the forms
N = — {2TrmkTy^^Uj/2
THE QUANTUM STATISTICS
401
PF = ft/ = ^ {ii^mkTY^krU ,,, . (263)
d) Degenerate case , — For this case of A large wc can obtain an
asymptotic evaluation of the integral by an application of
Sommerfeld’s lemma. We write
jj — I
r{v + 2)
du d
I ,, , du
- + I
Then by the lemma we easily find that
_ (log A)"+^ r , r (v + I)v
r(i.+ 2) (logA)^
-1- ^ (» + i)»(» - i)(^ - 2) , 1
+ (log Ay-
In particular,
u^/2 = —7- (log I +
8(I()g A)
Using (266) and (267), we find that our present expansions for N,
PV, and U are equivalent to (204), (ig8), and (205) for the case
where x is small. The connection between x and our present log A
is readily seen to be (cf. Eqs. [ry^], [lysK [194], and [260])
log A = \{x^ + i)‘^^ — i\dmc^ . (268)
b) Nondegenerate case. If A « 1, we can exi)and (r + Ac~") in
a series and evaluate U^, by integrating term by term. We find that
I Ca
Uo = r;y--~r~--<: I WV/w ^ ' A V (269)
* T- i; Jo
_ . _ ^ 4 _
“ ^ 2v-\-i ■» l-r 1-1 ” 1 “
402
STUDY OF STELLAR STRUCTURE
Equations (262), (263), and (270) now give deviations from
Boyle’s law, etc., owing to the exclusion principle even for “ordi-
nary” densities.
This completes the analysis of the gas laws, which should be
valid for an assembly of particles obeying the Pauli principle,
though the discussion has been carried through explicitly only for
the case of an electron gas.
10. The vibrations of the normal modes of a radiation field , — In
order to consider a radiation field in a manner analogous to the
treatment of an assembly of similar particles, it is first necessary to
find suitable co-ordinates to describe its motion. We have to start,
then, by an analysis of the number of possible modes of vibration in
a given frequency interval; this stage of the analysis corresponds
exactly to the discussion in § 2 of the number of independent eigen-
functions in a given energy interval.
Let xp stand for any one of the components of the electric vector
£ or the magnetic vector H, Then, according to the electromagnetic
theory, we have
Further, we have
(271)
div £ =
dx dy dz
o .
(272)
Let us consider for simplicity an inclosurc of the shape of a rec-
tangular box,
(273)
Let i/o he the value of ^ for a given t = o . We shall assume that
we can expand \po as a multiple Fourier series of the form
k^=o k,.=o jfe,=o
COS kxTTX COS kyTy cos k^TTZ
sin sin gi,^
(274)
THE QUANTUM STATISTICS
403
Similarly, if is the value of 3 ^/ 3 / at f = o, we can write
'V' A ' ^v’^y cos AjTZ ,
From (274) and (275) it follows that the solution of (271) is
_=o ^
COS kx'KX COS kyiry cos kz'KZ
sin \x Sin \y sin ^
(276)
where
v{|+|+|}, (,„)
and V is the frequency of the radiation considered.
It is clear that in each of the expressions (274), (275), and (276)
there arc eight possible terms and eight independent coefficients,
A and A', for given ky, and k,.
We must now consider the boundary conditions more closely. We
assume that the walls of the inclosurc are perfect conductors, so that
if ^ = Ex, then E^ should vanish on the two walls parallel to the
(iV) ") plane , that is. Ex = o at y = o, y = /y, 2 = o, s = /*, which
leaves only two terms of the type
COS kxTTX
sin lx
sin
kywy
sin
(278)
Similarly, the Fourier expansions for Ey and E, must contain re
spectivcly only terms of the types
and
. kxTTX cos kyTcy .
sin — — . sin
lx Sm ly
kjKZ
l 7
. kxTX .
Sin — ; — Sin
kyiry cos kxTZ
ly sin Ij,
(279)
(280)
404
STUDY OF STELLAR STRUCTURE
Since, however, the div E must vanish, the pure sine terms are im-
possible. Of the one remaining term in each of £*, Ey, Eg, only two
remain independent when (272) is satisfied. Thus there are two
normal modes of vibrations for a given set of values kx, ky, and kg
which satisfy (277).
Let us return to equation (277), which we can write in the form
where
•^x I 111/ I
.-2 ^2 ”T"
(281)
ax
2V
Qy ly ,
(282)
The number of normal modes of vibration with frequencies v ^ Vo
is equal to twice the number of points with integral co-ordinates in-
side an octanU of the ellipsoid (281) with v = Vo, which has the
volume
yC
(283)
By (282) and C283) we thus have for the number of normal modes
of vibrations in a radiation field with frequency ^ Vo
I 47r
2 ^ ■ O'xOfyQ'g
o 3
^vUxlyl..
(284)
Hence, the number of normal modes of vibration with frequcMicies
between v and v dv is given by
SttF
v^dv ,
(285)
where we have replaced IJLylg by V , the volume of the inclosure.
Actually, the result has been obtained for an inclosure of a rectangu-
lar shape, but a somewhat more comprehensive analysis by Weyl
shows that the result is completely general.
It is of interest to notice that if, in the expression (64), which gives
^ Only an octant, since kx, ky, and ks are by definition (Kq. [274]) nonnegative.
THE QUANTUM STATISTICS
405
the number of quantum states for a particle of mass m and energy
E in the range E, E + AE, we put
E = hv and m = o , (286)
we obtain precisely the expression (285).**
II. The statistics of light quanta. To be able to apply the laws of
statistical mechanics to a field of radiation we first recall that, ac-
cording to the quantum theory, each active mode of vibration with
frequency v is associated with an energy hv of the field.
With some slight modifications the Pauli-Gibbs theory given in
§ 3 is capable of handling the present case.
Let us consider a field of radiation in a given volume V and with
an internal energy U due to the active normal modes of vibration.
A microscopic state of the radiation field will be completely deter-
mined by the specification of the number of active modes of vibra-
tion with a frequency and energy //^a. We then have
U = . (287)
.V
A iK)ssil)le sequence of numbers w,, //j, . . . . , . . . . , must satisfy
(287). We shall write the dilTerent sequences of values for the w.,’s
which satisfy (287) in the form
//,D)
7 // ’ ^ 1
• • 1 '*'« > • • • • J
^‘0 )
• •
• • > »••••)
77 * * ^
0 j
^ , . .
( 1 )
' • J « J • * • * 1
(288)
“0 >
Hi J . ,
where W is the number of dilTerent solutions in integers of (287).
The entroj)y, S, of the radiation field is now defined by (cf., Eq.
[75I)
= W . (289)
^ 'this shows ji certain formal cquivji.lcncc ( from the present point of view) of lif?ht
(|uantii and a particle of zero rest mass which satisfies the Dirac equation.
4o6
STUDY OF STELLAR STRUCTURE
As in the discussion in § 3 we now drop the restriction (287) on
the w«’s by the passage to a canonical state where the energy U is
no longer defined exactly but is distributed in such a way that U has
a sharp maximum at a certain prescribed value say U,
According to (287) and (289), we now have
^Slk-dU — ppTg 8 ^ (290)
where 1? is, for the present, an arbitrary constant. Equation (290)
can be written more “symmetrically” in the form (cf. Eqs. [78] and
[80])
gS/k-dU — » . (291)
i
We now drop the restriction (287) and write, instead of (291),
^S/k-dU
(292)
where the summation over the nsS is taken over all the possible w/s.
But I? is now so chosen that
I
k
(293)
or, exactly as in § 3, = i/kT. If F is the free energy of the radia-
tion field, equation (292) can now be written as
e-mr = n 2 e"''”'"-''"' • (294)
We can show, exactly as in § 3, that (293) and (294) define for U an
extremely sharp maximum at U = U (say), and that U is ap-
preciably different from zero only in the range U ± AU where
THE QUANTUM STATISTICS
407
where (AJ 7 )^ is used to denote the ‘^mean square deviation” from J 7 .
In the same way, we can show that the entropy, defined according
to (293) and (294), differs from that defined according to (289) only
by a quantity of the order (cf. Eq. [105])
(29s')
Finally, we remark that (294) is now interpreted by the state-
ment that the probability of a microscopic state defined by the se-
quence (fiij . . . . , fte, ... .) (which define the number of active
modes with frequencies . . . . , . . . .]) and an energy U =
is proportional to
s
= jj^ ^
s
(296)
111 order, then, to obtain the probability of a microscopic state
with a definite total energy U which corresponds to the canonical
distribution (296), we must sum over all the sequences [n^] which
lead to the energy U.
For the internal energy V , we have, immediately,
N njip„c
IJ =
If, as in (142), we now define
(297)
(298)
then we can write
4o8
STUDY OF STELLAR STRUCTURE
This solves the statistical problem, and to obtain explicit formulae
we have to evaluate a,. We shall assume that n, can take all values
from o to 00 . This means, according to the discussion of § 4, that
the wave functions which describe the radiation field should be sym-
metrical in all the normal modes, each normal mode for this purpose
being described as a simple harmonic oscillator. Hence,
or
(Ti = -
I
I
e-koJkT ’
Therefore, by (299),
(300)
(301)
(302)
We can replace the sum by an integral and, weighting each frequency
interval by the appropriate density of the normal modes specified
by (285), we have
C’ — 1
(.303)
It is clear that if we wish to find the energy in the radiation field
in a given frequency interval, then we have a sum similar to (297),
the summation now being extended only over the required frequency
interval. We thus have
u^lv =
dv
c 3
(304)
which is Planck’s law. Since the radiation is isotropic, the Planck
intensity, is related to u, by (Eq. [29], v)
or
tlf, ,
47r
2hv^ 1
— I ■
(30s)
(306)
This completes our discussion of the quantum statistics.
THE QUANTUM STATISTICS
409
BIBLIOGRAPHICAL NOTES
The following general references may be noted.
1. W. Gibbs, Elemmitary Principles of Statistical Mechanics , Yale, 1902.
2. P. and T. Ehrenfest, EncyklopUdie der Mathematischen Wissenschaften,
4, Part 32, 1911.
3. P. Jordan, Statistische Mechanik auf qua^itentheoretischer Grundlage,
Braunschweig, 1933.
4. R. H. Fowler, Statistical Mechanics^ 2d ed., Cambridge, 1936.
5. H. A. Lorentz, Lectures Theoretical Physics^ 2, 141-188. New York:
Macmillan, 1927.
A general account of the physics of matter at high temperatures and densities
is given by —
6. F. Hund, Erg. exakt. Nalurwiss.y ig, 189, 1936.
The following particular references may be given.
§ I. — Ref. 4, chap, xvi; also,
7. R. H. Fowler, M.A., 87, 114, 1926, where the law p = A^,ps/3 is de-
rived and the first astronomical application given.
8. E. C. Stoner, Phil. Mag.^ 7, 63, 1929.
9. W. Anderson, Zs.f. Phys., 54, 433, 1929. Anderson was the first to rec-
ognize the importance of the relativistic effects in astronomical applications.
See also
TO. W. Anderson, Tartu Pub., 29, 1936,
1 1. E. C. Stoner, Phil. Mag., 9, 944, r93o, where the correct expression
for the internal energy, U, for a completely degenerate electron gas is obtained
for the first time.
12. S. ChlANDKASEKUAR, M.IV., 9I, 446, 1031.
13. S. CirANDRASEKiiAR, Ap. J 74, «Si, 1931. Ill references 12 and 13 the
law p = is used for the first time.
14. L. Landau, Phys. Zs. d. Soviet Union, i, 285, 1932.
15. T. K. Stekne, M.N., 93, 73^> ^933*
'I'hc following reference (in which the law p = is implicitly contained)
may be noted.
16. J. Frenkel, Zs. f. Phys., 50, 234, 1928.
§ 2. — 'I'he number of normal modes of vibration, when relativity effects are
taken into account, was first derived by -
17. P. A. M. Dirac, Proc. R. Soc., A, 112, 660, 1926 (the unnumbered equa-
tion on page 671 of this paper). For the corresiK)nding unrelativistic treatment
see Fowler, ref. 4, chap. ii. See also
18. . 10 . Ficrmi, Zs.f. Phys., 36, 902, 1926.
19. J. VON Neumann, Zs. f. Phys., 48, 868, 1928.
20. C. G. Darwin, Proc. R. Soc., A, 118, 654, 1928.
2r. C. Miller and S. Chandrasekhar, M.JV., 95, 673, 1935.
410
STUDY OF STELLAR STRUCTURE
22. R. Peeerls, M,N., 96, 780, 1936. This paper contains the most general
derivation of the expression for the number of normal modes.
23. E. K. Brock, Phys. Rev., 51, 586, 1937, where the Dirac equation in a
spherical potential hole is solved exactly and the enumeration of states is rigor-
ously carried out.
Also Jordan, ref. 3, chap, ii, §§ i and 2.
It has recently been contended by Eddington {M.N., 95, 194, 1935) that
the theory of a relativistic gas based on the expression for Z(E)dE derived in
this section (Eq. [66]) is incorrect. However, the investigations (refs. 21, 22,
and 23), undertaken, incidentally, also with a view of examining Eddington’s
contention, have failed to support it. The general theory presented in this
chapter is accepted by theoretical physicists (cf. Hund, ref. 6).
§ 3. — ^The account given here follows closely:
24. W. Pauli, Zs . f. Phys., 41, 81, 1926. Also ref. i.
§ 4. — R. H. Fowler, ref. 4, chap, ii; also,
25. P. A. M. Dirac, The Principles of Quantum Mechanics, 2d ed., chap, x,
Oxford, 1935.
26. A. SOMMERFELD, Zs. f. PhyS., 47, I, 1928.
§5. — ^The fundamental equations in the forms given in equations (170),
(171), and (172) are due to —
27. F. JuTTNER, Zs. f. Phys., 47, 542, 1928.
§ 6. — The lemma proved in this section is due to Sommerfeld (ref. 26). Gen-
erally, four cases have been distinguished: (a) unrelativistic nondegeneracy,
(6) relativistic nondegeneracy, (c) unrelativistic degeneracy, and, finally, (d) rel-
ativistic degeneracy. However, in the discussion we have distinguished between
degeneracy and nondegeneracy and taken account of the relativistic effects ac-
curately in both cases. The presentation which results is more elegant.
Also,
28. R. C. Maju2Hdar, Aslr. Nachr., 247, 217, 1932.
§ 7. — The nondegenerate case, including the relativistic effects, was first
considered by —
29. F. Juttner, Ann. d. Phys., 34, 856, 1911.
30. F. Juttner, Ann. d. Phys., 35, 145, 1911.
Also,
31. M. Planck, Ann. d. Phys., 26, i, 1908.
32. R. C. Tolman, Phil. Mag., 28, 583, 1914.
33. W. Pauli, Zs . f. Phys., 18, 272, 1923. The treatment given here of ther-
mal equilibrium between radiation and free electrons requires the use of the rela-
tivistic statistics.
THE QUANTUM STATISTICS 41 1
34. W. Pauli, ‘‘Relativitatstheorie,” Emyk. Math. Wiss., 5, Part 19,
641-674.
This section contains some unpublished investigations of the writer.
§ 8. — The analysis in this section is from Juttner, ref. 27.
§ 9. — Fermi, ref. 18; Sommerfeld, ref. 26.
§ 10. — Fowler, ref. 4, chap, iv; Pauli ref. 24; also,
35. H. Weyl, Math . Ann ., 71, 441, loir.
§ II. — Fowler, ref. 4, chap, iv, § 431.
The following references may also be noted:
36. R. C. Majumdar, A,N.^ 243, 5, 1931.
37. R. C. Majumdar, A.N.y 247, 217, 1932.
38. B. SWIRLES, M.N., 91, S61, 1Q31.
39. S. Chandrasekhar, Proc. R. Sac., A, 133, 241, 1931.
The foregoing papers deal with the problem of opacity of degenerate matter.
The following consider the various transport phenomena (conduction, viscosity,
etc.) .
40. D. S. Kotiiari, Phil. Ma^., 13, 361, 1Q32.
41. D. S. Kotiiari, M.N., 93, 61, 1032.
Also
42. 1 ). S. Kotiiari, Proc. R. Soc.^ A, 165, 4S6, 1938.
CHAPTER XI
DEGENERATE STELLAR CONFIGURATIONS AND
THE THEORY OF WHITE DWARFS
The white dwarf stars differ from those we have considered so far
in two fundamental respects. First, they are what might be called
“highly underluminous”; that is, judged with reference to an “aver-
age” star of the same mass, the white dwarf is much fainter. Thus,
the companion of Sirius, although it has a mass about equal to
that of the sun, is yet characterized by a value of L which is only
0.003 times that of the sun. Second, the white dwarfs are char-
acterized by exceedingly high values for the mean density; in fact,
we encounter densities of the order of 10^ and even 10^ gm cm“’. It
is this second characteristic which is generally emphasized, though
from a theoretical point of view the fact that L/Lq is generally very
small is of equal importance.
Since the radius of a white dwarf is very much smaller than that
of a star on the main series, it follows that for a given effective
temperature the white dwarf will be much fainter than the star on
the main series. Similarly, for the same luminosity the white dwarf
will be characterized by a very much higher effective temperature
(i.e., much “whiter”) than the main-series star; this, incidentally,
explains the origin of the term “white dwarf.”
We shall discuss the observational material in somewhat greater
detail in § 3, but it should already appear plausible that the white
dwarfs differ from other stars in some fundamental way. The clue
to the understanding of the structure of these stars was discovered
by R. H. Fowler, who pointed out that the electron gas in the in-
terior of the white dwarfs must be highly degenerate in the sense
made precise in the last chapter. We shall see that the white dwarfs
can, in fact, be idealized to a high degree of approximation as com-
pletely degenerate configurations. In this chapter we shall be main-
ly concerned with the applications of the theory of degeneracy
toward the elucidation of the structure of the white dwarfs.
412
STELLAR CONFIGURATIONS AND WHITE DWARFS 413
I. The gaseous fringe of the white dwarfs.— It is clear that the
extreme outer layers of a white dwarf must, in any case, be gaseous,
i.e., nondegenerate, with the perfect gas law, p oc pT, obeyed.
The question then arises as to how far inward we can descend be-
fore degeneracy sets in. To answer this question we shall have to
consider the criterion for degeneracy which was established in the
last chapter (Eq. [211]) and which we shall now write in the form
f(x)
where
^ /(^) = ^(2^ - 3)ix‘ + -f- 3 sinh-' a: . (2)
Finally, x is related to the mean electron concentration, n, by (Eq.
[212], x)
Wc shall write
n
3^^
(3)
where
P =
3//-*
fiJI = Bx^ ,
(4)
B
= 9.82 X .
(s)
Anticipating our result that the region of the white dwarf where
tlie perfect gas law is valid is an outer fringe only, we can use for
describing the structure of this gaseous fringe the theory of the
stellar envelope given in chapter viii. On account of the very small
values of L and R for the white dwarfs, the quantity a as defined in
chapter viii (Eq. [54]) is very small indeed (i — j 3 ~ so that
wc can use the analysis of § 3 of chapter viii. Wc then have
and
4 uH GM (i \
' - TV T TT ({ - V
_ I - (i y
^ 30/(0; u V
( 6 )
(7)
414
STUDY OF STELLAR STRUCTURE
where p is the mean density, ? is the radius vector expressed in terms
of the radius of the star, and/(o; w*) is defined as in equation (55)
of chapter viii. Inserting numerical values and expressing L, M,
and R in solar units, we find that
T = S-43 X io«/i
M (i \
R U V
(8)
^3. 75 Jo. S V-'*
p (i - [lr^-^J U V ■
By (4) and (9), we now find
X 7-75X10 i) •
By (i) and (8) we find that
X J ^ (| - ■) ‘:?(5T7p: *■ ■ •
(9)
(10)
(ll)
From (10) and (i i) wc can determine the point at which the rij^ht-
hand side of (i i) is unity; at this point we may say that “degeneracy
sets in.”
For most practical purposes it is found that it is suflicient to con-
sider for fix) the limiting form which it takes for small values of x.
By equation (24) of chapter x
/Cv)'^g.vs (.\*“>o). ([2)
The inequality (11) now takes the simpler form,
Eliminating x between (lo) and (13), we find that
^ M.(i - U-R^'V U V
STELLAR CONFIGURATIONS AND WHITE DWARFS 415
Now, since for the white dwarfs L and R are quite small, it follows
that for values of f appreciably different from unity the right-hand
side is, in fact, much greater than unity. Thus, if we consider the
case of the companion of Sirius, for which (according to Kuiper)
Log M = — o.oi. Log L == — 2.52, and Log 72= — 1.7 1, equation
(14) takes the form
If we assume that jjl ^ iXe = i.o, A"o = /« = 10,^ then the right-
hand side of (15) is unity for ^ = 0.94. At this point, according to
Table 17, the mass traversed from the boundary is only 0.23 per
cent of the mass of the star; further, it is found that at this point
= 0.12, in agreement with our assumption that x is small. Final-
ly, at f = 0.94, according to (8) and (9), p is found to be 1730
gm cm“'^ while T is 1.7 X 10^ degrees. For some of the other white
dwarfs the situation is even more ‘‘favorable,” in the sense that the
gaseous fringe is of even smaller extent. We thus see that the ma-
terial of the white dwarf must be almost entirely degenerate; this
result is implicitly contained in Fowler’s work, but the arguments,
essentially in the form we have given them, are due to Stromgren
and Sidentopf.
2. Coml)lclcly dc^cncrale conjif'ifraiions. We have seen in § i that
the gaseous fringe of a white dwarf is of quite negligible extent, and
that, further, the radiation is entirely negligible indeed, in the
gaseous fringe i — ( 3 10 or less. It is almost certain (cf. the
discussion in § b) that in the interior i — (i does not exceed its
value in the gaseous fringe, and we are thus led to consider equi-
librium configurations which are completely degenerate and in
which the radiation pressure is entirely neglected. The general
theory given in this section is due to Chandrasekhar.
'J’he equation of state can be written as (cf. F2qs. [19], [20], and [21]
of the last chapter)
F = A fix) ; p = ntxjl = Bx ^ , (16)
* Accordinj; to Stromgren, under the conditions of the gaseous fringe of a white
dwarf, the guillotine factor h must be quite large.
4 i6
where
STUDY OF STELLAR STRUCTURE
. „ T> S>Tm?c^nJI o vx c / \
A = — rr- = 6.01 X lo” ; B = = 9.82 X icAfi, (17)
3*3
3*3
and
f{x) = x(2x^ — 3)(2? + i)'/’ + 3 sinh ~3 x .
The equation of equilibrium is (Eq. [6], iii)
(7?)= "4xGp.
Substituting for P and p according to (16), we have
From the definition oif(x) we easily verify that
df{x) __ Sx^ dx
dr {x^ + i)*/* dr ’
or
I df(x)
8x dx _ dVx^ + I
— — — Q
x^ dr + i)^/* dr dr
Hence, equation (20) can be rc-written as
+ I
±d_( ^ dVx^ + i \ ^ ttGB^
r^ dr \ dr ) 2/I *' *
Put
Then,
dr
y" = 0;=* + I .
r* dr \ dr ) 2A ^
{18)
(19)
(20)
(21)
(22)
(2.0
(24)
(25)
Let X take the value a:„ at the center. Further, let y„ be the cor-
responding value of y at the center. Introduce the new variables
ri and <l>, defined as follows:
r = OJ7 ;
y = yo<l>,
(26)
STELLAR CONFIGURATIONS AND WHITE DWARFS 417
where
yl = xl + 1 .
(27)
The differential equation finally takes the form
JL —
rj^ drj
(28)
By (26) we have to seek a solution of (28) such that <l> takes the
value unity at the origin. Further, it is clear that the derivative of
0 must vanish at the origin. The boundary is defined at the point
where the density vanishes; and this by (24) means that if rji speci-
fies the boundary, then
<I>(.V>) = • (29)
Jo
From our definitions of the various quantities it is easily seen that
where
(30)
(31)
specifies the central density. Also, we may notice that the scale of
length, a, introduced in (27), has, in terms of the natural constants,
the form
^ 47r;w/*,.//y„ \ 2 cg ) '
or, inserting numerical values.
a
7.71 X 10^
Jo
/iy„’ cm .
(33)
We shall now consider a little more closely the structure of the con-
figurations governed by the differential equation (28).
a) The polcntial. -The function </> has a physical meaning. If V
is the inner gravitational potential, then from the general theory
,(chap. iii, § 2)
I ilP^
p dr
dV
dr
(34)
4 i8
STUDY OF STELLAR STRUCTURE
From (i6), (i8), and (22) we see that
dV %A , V
iT 3?' (35)
or, integrating, we find that
V - —'^yo<t> + constant . (36)
If we choose the zero of the potential at infinity, we have by ( 29) that
the ‘‘constant” in (36) is [( 8 ^/J 5 ) — GM/R] (cf. Eq. [lo], iii). Hence,
V=-^y„U-r -
I \ GM
yof R
{r^R) (37)
b) The mass relation, — The mass, interior to a specified point 77,
is given by
M{ri) = 47rJ^ /c
By (30),
pi 7 *rfl 7 .
(38)
I
(39)
or, using the differential equation (28),
Substituting for a and po according to (27) and (31), we have
MU)= (41)
The mass of the whole configuration is given by
dri )
We notice that in (41) and (42) y© does not occur explicitly. It is,
of course, implicitly present inasmuch as y„ occurs in the differential
equation defining 0.
STELLAR CONFIGURATIONS AND WHITE DWARFS 419
c) The rdation between the mean and the central density . — Let
'p{ti) be the mean density of the material inside 17. Then
Miv) = ^Tra’rySpCiy) .
Comparing (40) aird (43), we have
p(^?) = yl 1 ^
po ^ iyl — n dri ■
(43)
(44)
From (44) we deduce that the relation between the mean and the
central density of the whole configuration is
where denotes the derivative of <^. It is of interest to notice the
similarity between the present relations (42) and (45) and the cor-
responding relations in the theory of polytropes (Eqs. [69] and [78]
of chap. iv).
d) An approximation for configurations with small central densities,
— By definition, yf, = xl + i, and we need a first-order approxima-
tion when xl is small. We shall neglect all quantities of order xi and
higher. Then,
y. = I + l-vf, .
Put
(46)
-\ = 0 .
yi
(47)
In our present approximation we have
<A = I - K-rS - 6 ) •
(48)
At the origin, <f> takes the value unity. Hence,
e(o) = A-il .
(49)
From (28) we derive the following dilTerential equation for 0 .
1 d^O I dd
2 drj^ 7} drj
(50)
420
STUDY OF STELLAR STRUCTURE
Finally, introduce the variable according to
f = 2^/-^ . (50O
Equation (50) now reduces to
which is the Lane-Emden equation with index « = 3/2, but the
solution we need is not the Lane-Emden function According to
(49), we need a solution of (51) which takes the value at ^ = o.
Now, according to the homology theorem of chapter iv, § 8, as ap-
plied to the case w = 3/2, if is a solution of (51), then
is also a solution, where C is an arbitrary real number. Hence, from
^3/2 we can derive a function satisfying (49) by a homologous trans-
formation of 03/2!
6 == ^0 ^3/2( • (s^)
Hence, by (48), (50), and (52)
4 > = 1 — ^xl{l — d3/2{V2Xo 7j)} + 0(xj) , (53)
which relates (j) with From (53) we see that for these configura-
tions the boundary rji must be such that
’?.) = o , (54)
since, according to (29) and (46), = y, 7 ‘ = i - If
is the boundary of the Lane-Emden function, then from (54) we de-
duce that
v/27o
(55)
Again, from (53) wc have
(It) — ^
( 5^0
Combining (55) and (56), we find that
(=(i(o.u2) '
( 57 )
STELLAR CONFIGURATIONS AND WHITE DWARFS 421
Further,
(1
\V
, /i ddj/A
(S8)
From ($8) and (45) we have
^ 3e',M ’
(S 9 )
which is precisely the relation between the mean and the central
density for a Lane-Emden polytrope of index n = 2/ 2. Again from
(42) and (57),
M - - 4 „ ('sV'" (p ■tin'
XttGJ \2 ) V d^ J
(60)
On the other hand, if — » o, we can write the equation of state ( 16)
in the form
i’ = y x* ; p = ^.v 3 , (61)
or
/’ = A'lps/J , (62)
where
A" = = 2 . 9.91 X lo'-*
^/js'/3 20 \7r/ fiy-'
(63)
Hence, configurations with small central densities (i.e., x., small)
are J^anc-Lmden polytropcs of index n = 3/ 2. '^I'he results based
o'l (f> 3 ) fhe theory of polytropes, and the approximation
derived from the exact dilTerenlial equation (28) for a-„-^o arc
easily seen to be equivalent. In particular, using (63), the mass re-
lation (60) can be re-written in the form
(64)
which is identical with the mass relation for a polytrope of index
n = 3/2 based on the law (62) (cf. Eq. [OpJ, iv).
c) The limiling mass. F’rom the dilTerential equation (28) we
see that
as
( 65 )
422
STUDY OF STELLAR STRUCTURE
But from (33) it follows that at the same time the radius tends to
zero. From the mass relation (42), on the other hand, we see that
the mass tends to a finite limit:
/2^\3/*I (j..ddA
\-kg) di)
i=St(0s)
( 66 )
The existence of this limiting mass was first isolated by Chandra-
sekhar, though its existence had been made apparent from earlier
considerations by Anderson and Stoner, who, however, did not con-
sider the problem from the point of view of hydrostatic equilibrium.
For Xo — » 00 we can write (16) in the form
p =
2 Ax^ ; p = Bx ^ ,
(67)
or
p = ii:.p 4/3 ,
(68)
where
3 y /3 he 1 . 231 X lo's
TT/
(69)
By equation (70) of chapter iv the mass of a Lane-Emdcn configura-
tion based on (68) is given by
M = —4T
(70)
which is seen to be equivalent to (66) on substituting for A\ accord-
ing to (69).
Wc shall denote by M3 the limiting mass (66).® The mass relation
(42) can then be written in the form
M(yo) = M3
Q(yo)
0W3
where
0CJ3 = —
(e
(71)
(72)
“ We denote the limiting mass by M3 since, asic— ► ®,0— > ^3, the Lane- limclcn func-
tion of index 3.
STELLAR CONFIGURATIONS AND WHITE DWARFS 423
As the mass of the configuration increases monotonically with in-
creasing ya, we have the useful inequality
Q(y„)
< (yo finite) . (73)
Finally, we may note that the insertion of numerical values in the
formula for yields
Afs = S- 7 SMr X G . (74)
/) The internal energy.— By equation (23) of chapter x, the in-
ternal energy U of the configuration is given by
U = A J^^{ 8 a: 3 [(i -f - i] - f{x) \dV ; (75)
or, using equations (i6) and (17) which express the equation of
state, we can re-write the foregoing in the form
gyf rR
^ p[(i + - i\dV - j PdV . (75')
But by equation (32) of chapter iii the second term on the right-
hand side of (75 ) is --fi/3 where il is the potential energy. Hence,
8-1
^ = i-Jo + ^^y^^ - ^]dM{r) + ; (76)
or, expressing x in terms of 0 (cf. Eqs. [24] and [27]), we have
Using (37) for expressing <f> in terms of the potential V, wc obtain
+ + ( 77 )
Finally, using equation (i6) of chapter iii, we find
R ■
(78)
424
STUDY OF STELLAR STRUCTURE
For the case under consideration the internal energy is due entirely
to the kinetic energies of the motions of the electrons; we can, there-
fore, write
r - U = - p — . (79)
The total energy, E, of the configuration is
E = u + a= ( 79 ')
For stars of small mass the configurations are (as we have shown in
section d, above) polytropes of index n = 3/2, and by equation
(90) of chapter iv,
6 GM^
(80)
By (79) and (80) we have
T = — afi , (80')
which is the statement of the virial theorem (chap, ii, § 10) derived
on the basis of Newtonian mechanics. On the other hand, if
M M3, then (again by Eq. [90], iv),
(81)
By (79) and (81) we now have
T= -U, (81O
which must be the statement of the virial theorem for material par-
ticles moving with very nearly the velocity of light.
g) General results . — In sections d and e we have considered cer-
tain limiting cases. However, the exact treatment on the basis of the
differential equation (28) will provide much more quantitative in-
formation. The boundary conditions.
«#>=!,
d(l>
drj
o
at
r} = o
STELLAR CONFIGURATIONS AND WHITE DWARFS 425
combined with a particular value for will determine ^ completely
and therefore the mass of the configuration as well. Equation (28)
does not admit of a homology constant, and hence each mass has a
density distribidion characteristic of itself which cannot he inferred from
the density distribution in a configuration of a different mass. This is
the most fundamental difference between our present configurations
and the polytropes. We thus see that each specified value for yo de-
termines uniquely the mass M, the radius J?, the ratio of the mean
to the central density, and the march of the density distribution.
We have (collecting our results):
M ^ njyo)
0W3
^ ^
“ yo '
§ = {y2 - i)’/' ,
£ ^ I a (^\
p» f
\ yfj
In (82) we have introduced the unit of length (/i = ay.),
a(ir™-x
which, therefore, does not involve the factor in y„. Further, the physi-
cal variables determining the structure of the configurations are:
426
STUDY OF STELLAR STRUCTURE
K) Numerical results . — In section g we reduced the problem of the
structure of degenerate gas spheres to a study of the function 0 for
different initially prescribed values of the parameter y^. The inte-
gration has been numerically effected by Chandrasekhar for ten dif-
ferent values of the parameter:
~ = 0.8, 0.6, 0.5, 0.4, 0.3, 0.2, o.i, 0.0s, 0.02, o.oi .
The integration is started at the origin by a series expansion and then
continued by standard numerical methods. The following expan-
sion for <t> near the origin may be noted here :
<t> =
2! + J! -4 _ gKsg^ + 14) 6 . g‘( 339 g* + 280)
6 ’ ^40’ 7! ^ 3X9! ^
_ g^CuaSg'* + ii 436 g" + 4256) _|_
S X ii! r , ^ ,
where 5^ = {yl — i)/yl. The important quantities of interest are the
boundary quantities occurring in equation (82). These are tabulated
in Table 25. From the figures in Table 25 it is easy to calculate the
TABLE 25
Thk Constants of the White-Dwarf Functions
1/^3
Vt
po/p
0
6.8968
2.0182
54 -
O.OI
5 -3571
I. 9321
26 . 203
0.02
4 9857
1.8652
21 .486
o.os
4.4601
I . 7096
16.018
O.I
4.0690
1.5186
12 .626
0.2
3.7271
I . 2430
9 . 934 «
0-3
3 5803
1.0337
8.6673
0.4
3 5245
0.8598
7.8886
0 -S
3 5330
0 . 7070
7.350s
0.6
3 6038
0.5679
6.9504
0.8
4.0446
0.3091
6.3814
1 .0
CO j
0
5.9907
mass in units of M3, the radius in units of /„ and the central density
in units of B (= 9.82 X lo^n, gm cm“’). These express the chief phys-
ical characteristics in the “natural system” of units occurring in the
theory of these configurations (see Table 26). In Table 27 they are
STELLAR CONFIGURATIONS AND WHITE DWARFS 427
converted into the more conventional system of units which express
the radius and the density in c.g.s. units and the mass in units of the
TABLE 26
1 'HE Pfiysical Characteristics of Completely degen-
erate Configuration in the “Natural*' Units
ihl
M/Mi
/?//.
p »/5
I
0
00
0 .01
0-95733
o.S 3 S 7 i:
985.038
0.02
0.92419
0.70508
343 -
0.05
0.84709
0.99732
82.8191
0.1
0.75243
I . 28674
27.
0.2
0.61589
1.66682
8.
0.3
0.51218
1.96102
3 56423
0.4
0.42600
2.22908
r. 83711
o-S
0.35033
2.49818
I
0.6
0.28137
2.79148
0.54433
0.8
0.15316
3.61 760
0.125
1,0
0
CD
0
'VMiLK 27*
The Piiysk'al Characteristics of Completely
Degenerate Confkiurations
^/yl
M/Q
po in Grams
per ('ubic
Centimeter
Pmean
per Cubic
Centimeter
Radius in
Centimeters
5 • 75
00
GO
0
O.OI •
5 • 5 «
().«SXio"
3.70X107
4. 13X10*
0.02
5-32
3 37X10*
I .57X107
5.44X10**
0.05
4.87
8.13X107
5.08X10“
7 .69X10**
4 • 33
2.65X107
2 . loX 10“
9.92X10**
3 • 54
7.85X10^
7.9 Xios
1 . 2qX 10“
0.3
2.95
3.50X io<^
4.04XIOS
1 .51X10’
2-45
I .80X lo^*
2 . 29X10S
I . 72X10’
2.02
9.82X10S
T .34X lOS
1 ,93X10’
1.62
5.34XIOS
7.7 Xio‘i
2.15X10’
0.8
0 . 88
I .23X lOS
1 .92X10*’
2.79X10’
1 .0
0
0
00
* The values Kivfii in this tjililc dilTcr slivhtly from the published values (S. ('handrasekhar M.N..
95, 2oH, 'ruble HI). 'I'he dilTereiice is <liie tci llu* change in the acceple<l value.s of the fundamcntul
physical constants. , , ,
The calculations are for /« . = i . I'or the other values of n^,, M should be multiplied by/x ^, R by
anil /)„ by /u . '
sun. 'I'o see the order of magnitude of the quantities involved, it is
of interest to point out that the mass 4.87 has a radius only
428 STUDY OF STELLAR STRUCTURE
slightiy larger than the radius of the earth, while the mass 0.957^3
has a radius considerably less than the radius of the earth. In Fig-
Fig. 31. — The solid-line curve represents the exact (mass, radius) relation for the
completely degenerate configurations. I'his curve tends asymptotically to the dotted
curve as M~^o.
urcs 31 and 32 we have illustrated the mass-radius and the mass-
central density relationships. The dotted curves in the two cases
are the corresponding relations based on the Lane-Emden poly trope
3 log X
STELLAR CONFIGURATIONS AND WHITE DWARFS 429
of index w = 3/2 (the approximation considered in section d, above),
and the exact curves tend toward these asymptotically for ilf — > o.
We notice from Figures 31 and 32 how marked the deviations of
the dotted curves from the exact curves become even for quite small
masses. Thus, for M = 0.15M3 the central density predicted by the
exact treatment is about 25 per cent greater and the radius about 5
per cent smaller. The relativistic effects on the equation of state
Fk;. 32. - 'Fhc solid-line curve represents the exact (miuss, J^og po) relation for the
completely degenerate configurations. 'I’his curve tcntls asymptotically to the dotted
curve as M —>0.
arc therefore quite significant even for small masses. They certainly
cannot be ignored for masses greater than o.2My Of course, the
extrapolation of the n = 3/2 configurations for masses approach-
ing is quite misleading. The completely degenerate configura-
tions have a natural limit, and our discussion based on the differen-
tial equation shows how this limit is reached.
i) The relative density distributions in the different configurations . —
Our main diagram (Fig. 33) now illustrates the relative density dis-
tributions in the configurations studied. Here we have plotted p/po
against rj/rit for the different masses for which we have numerical
results. The two limiting density-distributions specified by the Lane-
iptp^)
430 STUDY OF STELLAR STRUCTURE
Emden functions ^3/2 and ^3 are also shown (dotted) in the same dia-
gram. The density distributions specified by the differential equa-
Fig. 33— The relative density distributions in the completely (Ie«eneriitc cun-
ligurations. The upper dotted curve corresponds to the polytropic distribution
and the lower dotted curve to the polytropic distribution « = 3. The inner curves
represent the density distributions for i/y„=*=o.8, 0.6, 0.5, 0.4, 0.3, 0.2, o.i, 0.05, 0.02,
and 0.0 1, respectively.
tion (28) thus form a continuous family which covers the range speci-
fied by the polytropic distributions of indices 3/2 and 3.
3. The discussion of the observational material and of the theoretical
mass-radius relation . — We have already seen in § i that the gaseous
STELLAR CONFIGURATIONS AND WHITE DWARFS 431
fringe of the known white dwarfs can be neglected (in the first ap-
proximation) and that we can regard them (in the first approxima-
tion) as completely degenerate configurations. The theory developed
in § 2 can therefore be applied, as it stands, to the known white
dwarfs. A glance at Table 27 shows that the mean density, the mass,
and the radius of these degenerate configurations are all of the right
order of magnitude to provide the basis for the theoretical discussion
of the structure of the white dwarfs. However, a really satisfactory
test of the theory will consist in providing an observational basis for
the existence of a mass such that as we approach it the mean density
increases several times, even for a slight increase in the mass. At the
present time there is just one case which seems to support this aspect
of the theoretical prediction.
The case in question is Kuiper’s white dwarf (AC 7o°8247), which
is, from several points of view, a most remarkable star; for instance —
and this is very unfortunate— in this star no spectral lines have been
detected so far and only a pure continuous spectrum has been ob-
served. According to Kuiper, the most probable values of L and
R are
Log Z,= -1.76, LogR=-2.38, (8s)
L and R being expressed in solar units. From (85) we derive that
p = 19,600,000 ^^^gm cm~’ . (86)
It is seen that we have here an unusually dense star. If we assume
that Hr = 1 .48, then the mass-radius relation established in § 2 leads
to a mass of 2.50, which would correspond to an actual mean den-
sity of 49,000,000 gm cm~’. On the other hand, if we use the approxi-
mation P = (Eqs. [62] and [63]), then from the mass-radius
relation for the poly tropes (Eq. [74], iv) we easily derive that
Log R = - J Log M - H Log fi„- 1.397, (87)
where R and M are expressed in solar units. Assuming = 2.0
(which is the maximum we can permit), we find that (87) leads to
a mass of 28 O for Kuiper’s white dwarf; it should be noticed that
432
STUDY OF STELLAR STRUCTURE
this is the minimum mass predicted on the basis of (87). (If we as-
sume for jLtfl the more probable value of 1.5, then (87) leads to
M = 118O.) Since the mass predicted on the model P ^ pS/3 comes
out unusually high, it seems likely that Kuiper’s white dwarf does,
in fact, provide a confirmation of the theory. In any case, it is clear
that if spectral lines could be detected and identified in this star and
the red shift measured, we might have a most valuable astronomical
confirmation of the physical theory of degeneracy.-^
However, since the theory is such a straightforward consequence
of the quantum mechanics and, further, uses Dirac’s theory of the
electron only in that phase of its application which has been con-
firmed by laboratory experiments (Klein-Nishina formula, produc-
tion of cosmic ray showers, etc.), there can be little doubt that it is
essentially correct.
We have seen that the theory provides a unique mass-radius rela-
tion if the radius is measured in units of h (Eq. [82]) and the mass
in units of But these units involve the ^'molecular weight,” ju,.,
so that we can apply the theory to determine p,* for white dwarfs for
which both M and R are known, or to determine M for a white
dwarf for which only the radius is known (assuming, however, a
value for It should be noticed that is not the same as the
mean molecular weight /x used in the theory of gaseous stars. For, as
the definition of /x,, we have used
p = npcH , (88)
where n is the number of electrons per unit volume. For a mixture
of elements which are all completely ionized we can write, in the
notation of § 3 of chapter vii,
n =
P
Az ’
(89)
where the element of atomic number Z and atomic weight yl / is
assumed to occur with an abundance xz by weight. The summa-
There is a possibility that Wolf 219, another white dwarf discovered by Kuiper,
for which Humason found recently a continuous spectrum, may be C()mi)aral)le lo
AC 7 o° 8247. If confirmed, this star would be even more extraordinary than AC 7o"S247,
since it is of lower luminosity.
STELLAR CONFIGURATIONS AND WHITE DWARFS
433
tion in (89) is extended over the elements present. Comparing (88)
and (89), we derive
=
(90)
If Xo is the abundance of hydrogen, we can re-write (90) as
I
= ^0 + 2
Z^i
xzZ
Az •
(91)
As a first approximation we can write ZjAz— 1/2 for all the
and obtain
/iff =
2
I + •
(92)
For the Russell mixture considered in chapter vii we find that
'*'■ 0.492 + 0.S08X,. • ^93)
We shall now consider brielly the other white dwarfs lor which we
have data.
a) Sirius B . — We have already considered this star in § i. Using
the data given there and using the theoretical mass radius relation,
it is found that y.,. = i.,^2, A'„ = 0.52.
b) 02 Eridani jB.- According to Kuiper,
Log L=— 2.26, Log M=— 0.3s, Log J?=— 1.74. (94)
I'he mean density is 91,000 gm cm“\ 'I'he theoretical mass-radius
relation leads to A'„ = 0.15.
c) Van Maanen No. 2.- -From the reliably known parallax and
spectral type, Kuiper derives for this star
Log L = -3.85, Log R= -2.05. (95)
'I'hc radial velocity of this star has been determined and found to
be +238 km/sec. According to Oort, most of this must be due to
the Einstein gravitational red shift. Assuming that the full amount
434
STUDY OF STELLAR STRUCTURE
is due to the red shift (which will give the right order of magnitude),
it is found, with the value of R given (Eq. [95]), that
Log M = o. 53 , p = 6,800,000 gm . (96)
The mass-radius relation now leads to = 1.206, Xo = 0.66.
4. A stellar criterion for degeneracy, — In the last chapter we
showed that the criterion for the applicability of the degeneracy
formulae is (Eq. [211], x).
x(i +
f{x) ^
(97)
However, for applications to stellar problems it is more convenient
to state the criterion for degeneracy in a rather different form.
Consider an assembly of N electrons contained in a volume V at
temperature T. Then, on the basis of the perfect gas law, the elec-
tron pressure pe would be given by
(98)
At temperature T we also have radiation pressure of amount given
by the Stefan-Boltzmann law
pr = \aT^ . (qS')
Let us denote by P the total pressure { = pr + p,) and introduce a
parameter defined as follows:
P = pr+ pe = — pe =
Pe
Pr-
Eliminating T between the relations (gg), we find
pe =
.
where we have used n for (N/V). Let
Shi
J
(00)
(too)
(lOl)
STELLAR CONFIGURATIONS AND WHITE DWARFS 435
as in equation (3). Then (100) can be transformed into
Pe =
/ ^i2irk* I — jS.V''’
3*3 \ | 3 « /
2X^ .
(102)
Since the radiation constant a can be expressed in terms of the other
natural constants as (Eq. [107], v)
8
^ 15 ’
equation (102) can be simplified to
(103)
p<, = A
/960 I — / 8 „y /3
| 3 - /
2X^ ,
(104)
where A is defined as in (17). It must, of course, be understood that
(104) is simply another form of (98).
Now for an assembly having the same number N of electrons in
the volume V, wc can formally calculate the electron pressure that
would be given by the degenerate formula, namely.
^•leg — Af{x) .
(los)
We have already shown (Eq. [26], x) that for all finite values of x
M
2X4
< I
(x < co) . (106)
Hence, comparing (104) and (105), we have the result that if for
a prescribed N and T, the value of / 3 „, calculated on the basis of the
perfect gas equation (98), be such that
960 I — / 3 r ^
(107)
then the pressure given by the perfect gas formula is greater than
that given by the degenerate formula not only for the prescribed
N and T, but for all values of N and T which specify the same / 3 „.
Let be such that
436
STUDY OF STELLAR STRUCTURE
or
^ ~ ~ 0-09212 . . . . ; = 0.90788 (109)
We can state the result just obtained in the following alternative
form. If for material at density p and temperature T the fraction
Y - ft), calculated according to {98), {g8'), and (pp), is greater than
ft), then the system is definitely not degenerate.
On the other hand, if
or
960 I - ft
ft
I - ft < I - / 3 „ ;
< I ,
ft > ft ,
(no)
(in)
then for the specified ft the electron assembly becomes degenerate
for sufficiently high electron concentrations. The criterion for de-
generacy under these circumstances would then be the following
For the specified N and T, calculate on the perfect gas law
(i.e., p, — nJiT) and solve the equation
/9^ I - /3. y/3 _ f{x)
\ ’T-' ft / 2X* '
(112)
(A solution exists, since [no] holds.) Denote the solution by a-' If
* for the prescribed N (according to Eq. [loi]) is much less than x',
then the system is far removed from degeneracy, while if a: is much
greater than x' the system will be more or less completely degen-
erate*
Table 28 provides solutions of (112) for different values of i -
If (no) holds, we can use the following approximation for the
real equation of state:
and
P.: = Af{x)
P.= 2a(^-^
ftV/-’
— X
x' being such that
\ ’T'* ft /
X*
(x ^ x')
(x ^ x') ,
( 960 1 - ft y/^ _ f(x')
\ ’T-' ft / 2X'‘>
(1I3)
(ii4)
STELLAR CONFIGURATIONS AND WHITE DWARFS 437
5. The effect of radiation pressure. The mass 21? = MjjS”*'''. — In
§ 2 we considered the equilibrium of completely degenerate con-
figurations, neglecting the radiation pressure entirely. This was jus-
tified in § I, where it was shown that for the known white dwarfs
these assumptions (of complete degeneracy and zero radiation pres-
sure) were entirely justified and our object in the study of the com-
pletely degenerate configurations is primarily one of obtaining a
satisfactory theory for the white dwarfs. It is, however, of some
theoretical interest to consider the effect of “introducing” radiation
pressure in these configurations.
Let us, in the first instance, consider a degenerate configuration
which is built on the standard model. Then the total pressure, P,
will be given by
P = ^7'po, (iiS)
where is the electron pressure and j8„ is a constant. Then, accord-
ing to equation (16),
F = /37M/(.r); p = 73 a- 3. (116)
It is clear that the analysis of § 2 applies to our present models if
we replace A (wherever it occurs) by 187 'yl. In particular, the mass
relation (42) now takes the form
M (/3„; y„) = — 47r
(117)
where ^ is, as before, a solution of (28). We can also write (117) in
the form
Af(/ 3 „.-y..) = yl 7 (i;y„)/ 3 , 7 ’'% (118)
in an obvious notation. In particular.
MiP,..; ^) = . (119)
From (t 18) and (i ig) it would at first sight appear that by allowing
/3«— » o wc can obtain degenerate configurations for any mass. This
is, however, incorrect. For, according to the criterion of degeneracy
established in § 4, (3„ has to be greater than |8„ if the matter is to be
438 STUDY OF STELLAR STRUCTURE
regarded as degenerate, and we see that the maximum mass of the
configurations which can be regarded as degenerate is therefore
given by
3)? = . (l 2 o)
The result just stated is extremely general and can be proved as
follows: Consider a completely degenerate configuration of mass M,
slightly less than My The density will everywhere be so great that
we can increase the radiation pressure from zero to a value only
slightly less than (i — at each point of the configuration and
still regard the matter as degenerate. According to (ii8), the mass
of the new configuration so obtained will be approximately M^^^
When M My the result becomes exact. We have thus proved
that the maximum mass of a stellar configuration which, consistent
with the physics of degenerate matter, can he regarded as wholly de-
generate, is SO? =
We may notice that
SD? = i.iSdAfj = 6.6sO/ir». (121)
6. Composite configurations. shall now give some elementary
considerations concerning stellar configurations with degenerate
cores, a subject initiated by Milne. Milne, however, considered de-
generate cores at such densities that the approximation P =
could be made. Since the exact treatment based on the differential
equation (28) leads to the existence of the two masses M,, and iW, and
since, further, there are no analogues to these on the approximate
considerations, it is clear that very considerable care should be ex-
ercised in interpreting the results derived on the basis of the approxi-
inate considerations. In particular, the formal results which are de-
rived for masses greater than 9JJ have no physical meaning. On the
other hand, it is possible to indicate the general characteristics of
these composite configurations by allowing the degenerate core to
be described by <l> without any elaborate machinery.
First of all, it is important to bear in mind that, while in the de-
generate regions the electrons contribute toward the pressure almost
entirely , the situation is different in the gaseous region : depending
on the abundance of hydrogen, the atomic nuclei would also con-
STELLAR CONFIGURATIONS AND WHITE DWARFS 439
tribute appreciably toward the gas pressure. The consideration of
the composite configurations which allow for these factors is ele-
mentary but complicated. However, the essential features of the sit-
uation can be understood by considering the case where we can put
lie = this implies that i — / 3 c = i — jS.
According to (104), we have (for the case under consideration) in
the gaseous region
P = (122)
and
p = BxK (123)
Eliminating x between (122) and (123), we have
P = 2a(^-^
\ ir^ /34
(124)
Wc shall assume that the gaseous region is governed by the stand-
ard-model equations, i.c., jS is constant in (124). The gaseous region
must then be governed by a solution d(^) of the Lane-Emden equa-
tion of index 3 -not necessarily 0 ^. 'Fhe mass relation (Eq. [70], iv)
is now
M
= — 47 r
/ 2/1 Y’A’ I /960 I — d0\
[itG) ) V
(125)
which by (66) can be written as
M
/ 960 I — 0)^
\ ‘ / c,W., '
(126)
where, in the notation of chapter iv,
0^3 =
W3
(e .
(127)
If the configuration is wholly gaseous, wc have
M = Mj
/960 I —
V ) ’
(128)
which is Eddington’s quartic equation in a different form.
440
STUDY OF STELLAR STRUCTURE
Now for a given mass M, equation (128) determines a jS =
Start with this mass having an infinite radius and imagine it being
slowly contracted. At first the configuration will be so rarefied that
it win be wholly gaseous and the path of the “representative point”
in the {R, i — / 3 ) plane will be along the line parallel to the A’-axis
through |3 = j8(lf). How far is this process of contraction possible?
From our criterion of degeneracy we can now conclude that if
I — jSfilf) > I — |8„, then the process of contraction is theoretical-
ly possible to an unlimited extent. Since j 3 „, according to definition,
is given by
960 I — fiu
IT*
( • -’9)
it follows that a configuration for which |8(Af) = / 3 „ is, according
to (128),
(..,0)
a) The domain of degeneracy . — For configurations of mass greater
than 2K, the appropriate i - p{M) is greater than (1 - and
the representative point will travel down the straight line (i = pi M),
however far the contraction may proceed. But the situation is dif-
ferent when the mass of the configuration is less than 9.)J. b'or such
masses, i — P(M) < i — and hence a stage must be reaihcd
when the configuration should begin to develop central rc'gions of
degeneracy. On the scheme of appro.ximation ( 1 1,0 and (i 1.4 1, we
can now easily see how far the process of contraction is possible be-
fore degeneracy sets in.
Let the central density be p„. Then
Po = Bxi.
Degeneracy would just begin to develop at the center for a value of
X = Xo such that
f(Xo) _ / 960 I -
2Xi \7r4 p ) ' ‘‘i’)
For this configuration the mean density p is (according to I-hp (yS],
iv, which gives the ratio of the mean to the central density fora
polytrope) <
- _ (1 do,
STELLAR CONFIGURATIONS AND WHITE DWARFS
441
The radius Ro of the configuration is, therefore, given by
ATfi _ Mass
* “ Mean density ' (^ 34 )
Substituting (125) and (133) in the foregoing expression, we obtain
Define a unit of length by (cf. Eqs. [27] and [33])
_ {2Ay/‘ 1,(^3) 7.71 X io» X 6.8g7 , ,,
~B > (^^36)
or, numerically,
^ = S -32 X loV”* cm .
From (135), then,
I \ T-t ) x„ ’
where afo is again determined from (132). By using (132), we can
write (138) more conveniently as
(137)
(138)
R
I
(I
(139)
It is a fairly simple matter to calculate from (132) and (139) cor-
responding pairs of values for {R„/l) and jS. 'These arc tabulated in
Table 28. This (R„, i — curve can therefore be drawn in the
(R, I — / 3 ) plane (see Fig. 34). The region bounded by this curve
and the two axes then defines the domain of degeneracy meaning
that it is only in this region that the curves of constant mass are
distorted from straight lines parallel to the 7?-axis.
From (132) and (139) we see that, as j8 ^ j8„.
a:,, — » 00 , Rb -^ o . (140)
Hence, as we should expect, the (R,,, i — 13) curve intersects the
(i — / 3 ) axis at a point where (3 = / 3 „. It can be proved easily that
the (R„, I — | 3 ) curve intersects the (i — jS) axis vertically.
b) The nature of the curves of constant mass for M ^ M, in the
442
STUDY OF STELLAR STRUCTURE
domain of degeneracy . — In (a), above, we have shown at what stage
a configuration of mass less than SD? (contracting from infinite ex-
(1-J8)-
Fig. 34. — The curve running from 1-/8=0.092 to infinity along the /\*-axis is
the (.Ro, I— /8) curve (see Kq. [139]). The points marked (5, .... , 15) on the (A*», 1 -(i)
curve and the iS-axis are the end-points (in the domain of degeneracy) of the curves of
constant mass for the values of M tabulated in Table 29. 'Fhc points marked ( i 4)
on the {Ro, i— /8) curve and on the (r— / 3 ) axis are the corresponding end points for
some curves of constant mass on the standard model (see d'able 30).
tension) begins to develop degeneracy at the center. "J'his happens
when the appropriate line (i — /8) = i — i 3 (M) intersects the
STELLAR CONFIGURATIONS AND WHITE DWARFS 443
(i?o, I — jS) curve. If the contraction continues further, the con-
figuration will begin to develop finite degenerate cores, and our prob-
lem now is to examine how the curves of constant mass run inside
the domain of degeneracy.
TABLE 28
The Stellar Criterion for Degeneracy and
THE ( Ro , I - /S) Curve
X
.-0
R./l
0
0
00
0.2
0 . 00040
I . 9868
0.4
.00282
1-.5787
0.6
.00793
I .0956
0.8
•01505
0.9187
1 .0
.02305
0.7934
1.2
.03101
0.6985
1-4
•03839
0.6235
1.6
.04495
0.5627
1-8
.05068
0.5123
2.0
•ossf’i
0 . 4699
2.2
• 0508.5
0.4337
2.4
.06344
0.4025
2.6
0.06653
0.3753
X
I -/S
Ro/l
2.8
0.06919
0-3515
3-0
.07149
• 3 v 304
3-5
•07508
. 2870
4.0
.07920
■2535
4-5
.08158
.2268
5-0
.08337
.2051
6.0
■08583
. 1721
7-0
.08739
. 14S1
8.0
. 08844
.1299
90
.08918
■ IIS 7
10.0
.08972
.1043
20.0
.09150
.0524
30.0
.09185
■0350
00
0.09212
0
In § 2 we made an analysis of completely degenerate configura-
tions. Each mass (less than M3) has a certain uniquely determined
radius, 'fbus, if the mass under consideration has a central density
corre.sponding to y = y,, then the radius, R, is given by
R = arfi
(141)
where 77 r is the boundary of the corresponding function <^(3/,,). In
terms of the unit of length, I (Eq. [136]),
^ ^ £ Vi [<t>{yn)]
(142)
These completely degenerate configurations correspond to jS = i.
Hence, we know from (142) the point at which the curves of con-
stant mass for M < M3 must intersect the ./^-axis. Also, for any
mass M we can calculate the value of jS in the wholly gaseous state.
Let jSf be the value of p for a wholly gaseous configuration which in
444
STUDY OF STELLAR STRUCTURE
its completely degenerate state has a central density corresponding
to y = yo. Then, according to equations (71) and (128), we have the
relation
960 I — jQt V''* _ _ tl(yo)
•tr* J Mi „wj ’
(143)
where, as in equation (72),
fl(y„) =
— ’ll
Av / lt=rii(<l>(yu))
(144)
Now the line through jSf parallel to the i?-axis will intersect the
(i?o, I — |3) curve at [i2o(M(y«)), i — jSfl. In the domain of de-
generacy the continuation of the curve must in some way connect
the point [R„{M(yo)), i — jSf] and the point R on the J?-axis, where
R I nMyojM))]
I yoiM) uos)
(14s)
From the numerical values for j;,, ft, etc., for the ten different
values of yo given in Table 25, the corresponding values of R/l
(according to [145]) and jSf can be evaluated. The results are given
TABLE 29
i/y;
M/Mj
I - fit
R/l
0
I .
0 . 07446
0
O.Ol
O.Q 5733
. 06966
0.07767
0.02
0.Q241Q
• 06596
0. 10223
0.0s
0 . 84709
.05746
0. 14460
0. T
0.75243
.04732
0. 18657
0.2
0.61589
• 033 S»
0. 241()8
0-3
0.51218
.02414
0.28434
0-4
0.42600
.01718
0.32320
o-s
0.35033
.01187
0.36222
0.6
0.28137
.00779
0.40475
0.8
o.iS 3 i^>
.00236
0.524.53
I.O
0
0
00
in Table 29. We have thus fixed the “end-points” for the curves of
constant mass for M ^ M3 in the domain of degeneracy. 'I'hc cor-
responding pairs of points on the (J?„, 1 — 0 ) curve and the R-axia
arc shown in Figure 34.
STELLAR CONFIGURATIONS AND WHITE DWARFS 445
It is clear that the curve for must pass through the origin of
our system of co-ordinates. Further, if jSo is the value of jS for M3
in the wholly gaseous state, then, according to (143),
960 I — ft
ir4 ft
(146)
or
i-ft = o.o7446; ft = 0.92554. (147)
c) The nature of the curves of constant mass for M > M3 in the
domain of degeneracy . — In (ft), above, the end-points for the curves
of constant mass (for configurations with mass less than, or equal
to, M3) have been fixed. We further saw that the curve for M3 must
pass through the origin. The question now arises: What happens
for configurations with ^ M > M3? The answer to this question
can be given quite simply if (i — jS) has the same value in the de-
generate core as in the gaseous envelope. We have already shown
(Eq. [119I) that the completely relativistic configuration has a mass
M = (1^/3^ ft) (148)
and is of zero radius. Hence, the curves of constant mass for M> Mj
must cross the (i — ft axis at a point (1 — / 3 *), say, such that
M = (149)
Let us denote by jSf the value of j8 in the wholly gaseous state.
There is a simple relation between jS* and /Sf. Comparing (143) and
(149), we derive that
=
(iL
\ 96 o I - /3t/
(150)
From (150) wc sec that 18*= i, /Sf = ft (Eq. [146]), is a solution;
in other words, the appropriate curve for M3 must pass through the
origin which in fact it does. Again, jS* = /Sj = ft is also a solution
of (150); the appropriate curve for is therefore the full line
through ( I — ft) parallel to the i^-axis, as we should have expected.
Table 30 gives a set of corresponding pairs of values for jS* and
446
STUDY OF STELLAR STRUCTURE
/Sf (see also Fig. 34, where the corresponding pairs of points are
marked [i, 2, 3, 4] on the [i2o, i — i 3 ] curve and the [i — jS] axis).
The results described above (in [6]) are true for the usual standard
model. If we consider as an another limiting case configurations in
which jS = I in the degenerate core and /S ^ i in the gaseous enve-
lope, then the discussion is similar but somewhat more complicated
(cf. Chandrasekhar’s papers quoted in the Bibliographical Notes).
TABLE 30
I -/3t
1
M/m
0.09212
0.09212
1 .
.090
.08220
0.9838
.085
.05768
0.9457
. 080
•03143
0.9075
.075
.00319
0.8692
0.07446
0
0.8651
A more detailed discussion of composite configurations would con-
sist in describing the mathematical methods for handling them pre-
cisely, i.e., by a consideration of the methods of fitting a solution of
the Lane-Emden equation of index 3 to a solution of the diiTcrential
equation for <^. Such discussions, however, are beyond the scope of
the monograph. Reference may be made to the literature quoted in
the Bibliographical Notes.
7. Partially degenerate configurations.- far we have considered
completely degenerate configurations and also stellar configurations
with degenerate cores. For describing the degenerate state we have
used the exact equation of state (allowing for relativistic elTecls)
which should be valid if the degeneracy criterion is satisfied. In
considering the composite configurations in § 6, we changed over
from the perfect gas equation to the degenerate equation of state
at a definite interface, the interface being defined in such a way that
both the equations of state give the same numerical value for the
pressure for the density and the temperature at the interface. We
have seen that this approximation is quite good so long as we deal
with configurations of not too small masses (in units of A/.,). How-
ever, for stars of small mass o. i the central density, even in the
completely degenerate state, is not unduly high. Under these cir-
STELLAR CONFIGURATIONS AND WHITE DWARFS 447
cumstances, we may expect that in actual stars (e.g., Kruger 6o) the
“transition zone” between the perfect gas region and the region of
more or less complete degeneracy will be quite extensive. It is there-
fore a matter of some importance to allow for these incipiently de-
generate regions in a satisfactory way.
We shall illustrate a method of approach to the problem just
stated in one case, namely, that in which the configuration is so poor
in hydrogen that we can put /x = /Xo = 2. Further, we shall assume
that the star is of such small mass that relativistic effects can be neg-
lected. Under these circumstances, the equation of state can be para-
metrically expressed as follows (cf. Eqs. [262] and [263], x):
Pm = Ji , ( 151 )
where IJ„ stands for the integral
U„ =
f oo
u^du
(152)
(153)
We shall assume that lUu known functions of A, so that
the parametric representation of the equation of state is in terms
of A.
We shall consider two classes of equilibrium configurations built
on the equation of state (151) and ( 1 52) which allows for the transi-
tion between p « pT to p ^ quite accurately.
a) The isothermal gas sphere. In this case, T' is assumed con-
stant, and the equation of equilibrium,
I d /r^ dp\
dr \p dr )
-47rGp ,
( 154 )
on inserting for p and p according to (151) and (152), becomes
T. Tr iw. ii
448
STUDY OF STELLAR STRUCTURE
Let
where
r = at ,
_ / U
“ \SirGi2Trmyi^{kT)^l^{nHy)
Equation (154) now reduces to
Now it is easily seen that
JL.U =
dA ro
r
u + i)A^ I
u“e“du I
Equation (158) can therefore be simplified to the form
? |(f w
If A «: I, we have (cf. Eq. [270], x)
U„ = A.
(161)
Hence, if we write
\ = e-* ; f = J ,
(162)
equation (160) transforms to
d^ dU
(163)
which is the isothermal equation of a classical perfect gas sphere
(cf. § 22, iv).
On the other hand, if A » i, then (Eq. [266], x)
U,/, = (log A) 3 /^ .
3 V TT
(164)
STELLAR CONFIGURATIONS AND WHITE DWARFS
449
Hence, if we make the substitutions
log A - « ; f ^ ^ ^ ,
(165)
equation (i6o) reduces to
i i. = _0,/.
(166)
which is the Lane-Emden equation of index n = 3/2. We thus see
that, depending upon T, wc obtain from (160) either the HassiVal
isothermal case «) or the poly trope, n = 2 >h (T—^o). A
closer study of the differential equation (160) than has yet been
made will make it possible to study how the change from the HassiVal
isothermal gas sphere to the poly trope n = 3/2 takes place as A
increases from very near 0 to <» . The discussion of (160) may lead
to results of cosmological importance.
b) The standard model. — ^We shall next consider the standard
model built on the equation of state, (151) and (152). Quite gen-
erally, on the standard model, we have
1 = 7 ^ •
Let
(167)
(). = |j ; (>. = 1 .
(168)
Equations (151), (152), and (1C7) can now be written as
P«uH = QMTy/^u,/. ,
(169)
p = Q,(kry/^ixn u.„ ,
and
(170)
II
X
(i7x)
From (t6q) and (171), we obtain
(172)
4S0 STUDY OF STELLAR STRUCTURE
Substituting for kT according to (172) in (169) and (170), we obtain
PP - (173)
and
p ~ QxQiixH Ui/ 2 U ^/2 • (174)
Substituting (173) and (174) in the equation of equilibrium (154),
we find
By ^159) we can simplify (175) somewhat into the form
?TrV ^b~) ; ■
Let
(17s)
(176)
(177)
Equation (176) now reduces to
If A <3C I, we have t/„ = A, and (178) can be written as
or, if
e = A«/3 ; f = Vij i ,
(r 78 )
(179)
(180)
equation (179) reduces to the Lane-Emden equation of index n — 3,
as would be expected.
On the other hand, if A » t, then, according to equations (266)
and (267) of chapter x,
S^TT
^(logA).^/-. (181)
U.,, =
(log A)5/“ ; t /,,/2 =
STELLAR CONFIGURATIONS AND WHITE DWARFS 451
If we now put
(log (.8,)
equation (178) reduces to the Lane-Emden equation of index n =
3/2. We thus see that a detailed study of (178), which has not as
yet been made, should give insight into the structure of partially
degenerate stars. The numerical discussion of the models (Eqs.
[160] and [178]) cannot be very difficult; a one-parametric series of
integrations would be sufficient.
BIBLIOGRAPHICAL NOTES
No attempt at a complete bibliography is made.
1. R. H. Fowler, M.N,y 87, 114, 1926. In this paper Fowler makes the
fundamental discovery that the electron assembly in the white dwarfs must be
degenerate in the sense of the Fermi-Dirac statistics.
2. E. C. Stoner, Phil. 7, 63, 1929.
3. E. C. Stoner, Phil. Mag.y 9, 944, 1930.
In references 2 and 3 Stoner makes some further applications of Fowler’s
idea. Milne and Chandrasekhar independently applied the theory of polytropes
to the case considered by Fowler:
4. E. A. Milne, M.N.y 90, 769, 1930.
5. S. CirANDRASEKUAR, Phil. II, 592, 1931.
Relativistic effects were first considered by- -
6. W. Anderson, Zs.J. Phys.j 54, 433, 1929, but the correct formulation of
Anderson’s work is due to Stoner (ref. 3). Relativistic degeneracy in conjunc-
tion with poly tropic tJieory was considered by C'handrasekhar:
7. S. CirANDKASEKUAR, M.IV.y 91,456, 1931.
8. S. Chandrasekhar, Ap. /., 74, Si, 1931.
The discussion of tlie theory of white dwarfs on the basis of the e.xact equa-
tion of state for a completely degenerate gas is due to —
9. S. Chandrasekhar, M.N.y 95, 207, 1935.
Stellar configurations with degenerate cores arc considered in —
10. S. Chandrasekhar, M.N.y 95, 226, 1935.
n. S. Chandrasekhar, M.N.y 95, 676, 1935.
Also,
12. S. Chandrasekhar, Ap., 5, 321, 1932.
13. P. TEN HrUOOENCATE, Ap.y II, 201, I936.
§ I. — 'rhe discussion given in this .section is due to -
14. IL Stromoren, Ihindb. d. J^hys.y 7, 160 61, 1936.
15. II. Siedentopk, A.lSf.y 243, I, 1931.
4S2
STUDY OF STELLAR STRUCTURE
§ 2. — In this section the analysis in reference 9 is reproduced.
§ 3. — The author is indebted to Dr. Kuiper for providing the observational
material. Kuiper’s discovery of his white dwarf is described in —
16. G. P Kuiper, Pub, AS.P,, 47, 307, 1935.
§§4, 5, 6. — The essential parts of the discussion in references 10, ii, and 12
are reproduced here.
§ 7. — This represents a hitherto unpublished investigation by the author.
The functions 27 , /a and 1/3/3 have recently been tabulated by E. C. Stoner:
17 E. C. Stoner, PMl. Trans. Roy. Soc. A., 237, 67, 1938.
CHAPTER XII
STELLAR ENERGY
In this chapter an attempt will be made to indicate some general
trends in the current approach to the difficult problem concerning'
the origin of stellar energy. This subject is as yet in an early stage
of development and the present brief alccount is intended primarily
to indicate the directions in which the greatest progress is being, or
is likely to be, made. This chapter, then, is on an entirely different
level from the preceding ones, in which an attempt toward rigorous
development has been made.*
I. The Helmholtz- Kelvin time scale,- We shall first examine the
reason for postulating a source of stellar energy. To see this, let us
consider gaseous configurations in which the radiation pressure can
be neglected; the majority of the normal stars (sun, Capella, etc.)
are in this category. Then, as we have shown from the virial theorem
in chapter ii, § lo,
E= (i)
where E, U, and il arc the total, the internal, and the potential
energies, respectively. If the configuration contracts and if, as a re-
sult of this, there is a change in the potential energy of amount dil,
then (as was shown) a fraction (37 — 4)/ 3(7 — i) of the energy
I AS 2 1 ‘liberated” is radiated to the space outside, while the remaining
fraction [i — (37 — 4)/ 3(7 — 0] is used in increasing the internal
energy of the configuration. Hence, the amount of energy, — AE,
which is radiated to the space outside, consequent to a decrease in
the potential energy of amount | AQ | , is given by
-AE= — Afl ■ (2)
3(7 - i) '
* This chapter was written in December, 1937, and consequently it has not been
possible to include the more recent investigatibns of Gamow, Bcthe, and others (see
the Bibliographical Notes at the end of the chapter).
453
4S4
STUDY OF STELLAR STRUCTURE
Now the contraction h3^othesis of the origin of stellar energy postu-
lates that the energy radiated by a star is due to a slow secular
contraction. (The contraction hypothesis is also referred to as the
“Helmholtz-Kelvin hypothesis.”) Thus, if the potential energy
alters by an amount Ai 2 in time A/, then the luminosity L is given by
T - - 37-4 ^ / X
A/ “ 3(7 -i) dt'
Now the contraction h3^othesis allows an estimate to be made of
the time during which the normal stars could have existed. To make
this estimate, let us suppose that the configuration was initially in an
infinitely extended state and that after a time t it has contracted to
a radius R and a potential energy fi. Then by (3) we should have
We can write
37-4
3(7 - 1 )
Q .
(4)
is)
where g' is a numerical constant of order unity; if the configuration is
a poly trope of index n, then (cf. Eq. [go], iv),
We can re-write (4) in the form
( 6 )
Lt
37 - 4
^3(7-1) R '
(7)
where L is the mean luminosity during the time t. Equation (7) al-
lows us to establish the time scale of Helmholtz and Kelvin. luir
the time during which a star with an observed luminosity L could
have existed while radiating at its present rate is given by
^H.K.
_ 37-4 GM ^
^ 3(7 - i) LR '
(8)
STELLAR ENERGY
455
If we assume 7 = 5/3, we find that the sun could have existed at its
present rate of radiating energy to the space outside for a time
not longer than
^H.K. (sun) = 1.59 X io 7 X years . (9)
If we assume that the sun has a polytropic density distribution of
index 3, then <7 = 1.5 and wc have
/h.k. (sun) = 2.4 X io 7 years . (10)
In the same way we find that
/h.k. (Capella) = 2.2 X 10^ years . (ii)
It should be pointed out that (10) and (ii) are not exact figures,
but it is clear that no reasonable adjustment of the parameters can
extend the time scale for the sun, for instance, to more than 10**
years. The order of the ^‘age” of the sun thus derived on the Helm-
holtz-Kelvin contraction hypothesis is found to conflict with other
evidence which is essentially of a geological nature. Thus, the age of
the terrestrial rocks as derived from the uranium-lead and helium-
lead ratios of radioactive minerals is in the neighborhood of 1.6 X 10’
years, and the sun must have existed in somewhat its present form
for at least this length of time. Hence, the geological evidence com-
pletely disproves the contraction hypothesis for the sun, and there-
fore also for the normal stars. Wc are thus led to seek a different
origin for the source of stellar energy.
2. Transniulalion of dements as the source of stdlar energy . — There
is also evidence in addition to that of a geological nature, which
points to an age for the sun (and therefore for the majority of the
normal stars) of at least 1 or 2 X years. Astronomical evidence,
the discussion of which is beyond the scope of the present mono-
graph, also points to a similar age (10'^- 10'" years). It should, how-
ever, be mentioned that this does not necessarily mean that every
single stellar objc'ct that is observed must have existed for this length
of time; it only means that some aspects of the stellar system (e.g.,
the rotation of the galaxy) could not have existed for a time much
longer than 10'" years.
4S6
STUDY OF STELLAR STRUCTURE
Now a source of stellar energy which will allow for most stars a
time scale of the order of years is the transmutation of elements
— a suggestion which appears to have been first seriously considered
by Harkins, Perrin, Eddington, and, more recently, by Atkinson and
Houtermans. As we have seen in chapter vii, hydrogen is abundant
in stellar interiors; and as we shall see in §3, the probability of
protons taking part in the transmutation processes is very much
greater than for the nuclei of the higher atomic numbers. Conse-
quently, most of the energy due to the transmutation processes will
arise from the building-up of the elements of higher atomic numbers
out of hydrogen. The mass of the hydrogen atom is 1.0081 in the
scale 0 = 16 and it is seen that the energy available from the trans-
mutation of a hydrogen atom is approximately 0.008 of its mass.
In other words, the energy available from a gram of hydrogen is
0.008 X = 7.2 X ergs . (12)
Now each gram of material from the sun liberates, on the average,
2 ergs per second. Hence, the order of the length of time during
which the sun (assumed to be initially a mass of pure hydrogen) can
go on radiating at its present rate before all the available hydrogen
is used up, is
J X 7. 2 X 10*® sec = I . I X 10“ years . (13)
Thus, on the transmutation hypothesis, the maximum time scale for
the sun is the “intermediate time scale.” If we consider the more
luminous stars, the time scale permitted will be very much more
limited; and unless we are willing to accept the hypothesis of the
annihilation of matter (for which hypothesis there is at the present
time no physical basis), we simply have to accept the much shorter
time scale for these stars. - In any case, we shall restrict ourselves to
a consideration of the transmutation of the elements as the source
of stellar energy. Further, we shall see (§§ 3 and 4) that, for the
order of the temperatures found for the stellar interiors (chaps, vi
and vii), transmutations by proton capture of the lighter elements
can take place at nonequilibrium rates. What is meant by a process
occurring at a “nonequilibrium rate” will be made more precise
STELLAR ENERGY
457
presently — but it may be mentioned here that some investigators
(Milne and Sterne) have considered the possibility of the transmu-
tations occurring at equilibrium rates. We shall not, however, con-
sider these theories, for, first, they require temperatures for the
stellar interiors of an altogether different order lo^ degrees or
more) from those for which we have any evidence; second (as
Stromgren has pointed out), if transmutations occurring at equi-
librium rates are to be regarded as the source of stellar energy, then
the Russell-Vogt theorem will not be applicable; but, as we have
seen (chap, vii), the observational material strongly suggests the
validity of the theorem; third, one of the main reasons for the con-
sideration of transmutations occurring at equilibrium rates arose
from the belief that stellar configurations built on the alternative
hypothesis (transmutations occurring at nonequilibrium rates) are
unstable, for which belief, however, there does not appear to be at
present any convincing reason. It is beyond the scope of the mono-
graph to go into greater details on these questions, but reference
may be made to a general discussion of these matters by Stromgren
in a recent article.^
3. The transparency of potential harriers. The Gamow factor , — We
have seen in § 2 that the most profitable approach to the problem
of the origin of stellar energy at the present time is made by examin-
ing the physical processes of the transmutation of the elements. For
example, a process which we shall consider is the disintegration of
lithium into two a-particles by the capture of a proton:
lLi+ \n^2lHe, (14)
The foregoing disintegration has of course been carried out in the
laboratory in the first experiments of Cockroft and Walton. We shall
presently see that transmutations of the kind (14) can — and do, in
fact- occur under the physical conditions which we have derived
for stellar interiors (chaps, vi and vii); we shall also sec that the
“reaction” converse to (14), namely,
2irie^lLi+]n , ( is )
“ B. Stromgren, Erg. exakt. Naturwiss., 16, 466, 1037 —especially §§ 14 and 18.
458
STUDY OF STELLAR STRUCTURE
does not occur. We are thus led to consider the transmutations of
elements occurring at nonequilibrium rates as the source of stellar
energy. This is different from assuming that reactions (14) and (15)
both occur at almost equal but slightly different rates, and that it is
the slight difference between the reaction rates in the two senses
that is responsible for the generation of stellar energy. This assump-
tion will have to be made if it is supposed that the transmutations
occur at equilibrium rates. However, for the reasons stated toward
the end of § 2 we shall not consider this alternative hypothesis.
If the transmutations occur at nonequilibrium rates, the problem
which we have to consider is the evaluation of the probability of the
penetration of potential barriers, which, on the basis of the classical
mechanics, cannot be expected to happen. The physical situation
can perhaps be understood by considering first the analogous prob-
lem of the a-decay of radioactive bodies.
Let us consider, for example, a uranium nucleus which is known
to be a active. From the experiments of Rutherford on the scatter-
ing of a-particles by the uranium nucleus it has been inferred that
the Coulomb law of attraction between the uranium nucleus and
the a-particle is valid up to a distance at least as small as 3 X
cm; this means that we have
V(r) = (>■ > 3 X 10“'=' cm) , (i6)
T
where we have written V{r) to denote the potential energy Ijetween
the two nuclei at the distance r. However, at much smaller distances
we should expect deviations from the Coulomb law, since the
stability of the uranium nucleus requires the existence of a “potential
hole” at the center of the nucleus. The general nature of the func-
tion V{r) must therefore be of the character shown in Figure 35, in
which the dotted line represents the Coulomb potential and the
solid line the actual potential. The inner part of the curve has l)een
drawn arbitrarily, but for r > 3 X 10“'^ cm the scattering exi)eri-
ments have shown that there is no appreciable deviation from
the Coulomb law. At the same time, the uranium nucleus, being a
active, emits a-particles which are found to have an energy of
STELLAR ENERGY
459
6.6 X lo”® ergs. It is, consequently, difficult to understand, on the
classical picture, how particles contained inside the potential hole
can go through a potential barrier which is at least twice as high as
their total energy. According to classical mechanics, particles with
energy 6.6 X lo”® ergs could originate only from a point at a dis-
tance of 6 X lo""" cm from the center, where the Coulomb potential
.^5. Potential cncri^y of an a-parlicic in the (IcUI of a uranium nucleus
has the value 6.6 X to”'* ergs. In this region, however, there can
be no question of the a-particle being stably bound to the rest of the
uranium nucleus. We thus see that apparently particles inside the
potential hole with energies much less than that corresponding to
the top of the potential barrier can, so to say, tunnel through. That
this paradox of the classical picture does not exist in the quantum
theory was, as is well known, shown to be the case by Gamow and
by C'ondon and Gurney.
'I'he possibility of a particle’s going through a potential barrier is
connected with the wave nature of the wave functions ^ and the
460
STUDY OF STELLAR STRUCTURE
interpretation of the square of the modulus of as the measure of
the density of the probability of the particle being in a certain region.
It is not true that the wave function vanishes in regions where the
potential energy V{r) is greater than the total energy £, and where
on the classical mechanics the particle will have a negative kinetic
energy. Actually, the wave function, although it decreases ex-
ponentially as we go out into the ‘‘forbidden region,^' is yet finite at
great distances, thus giving a finite probability that a particle may
appear in such regions, and, in particular, penetrate the potential
barrier. An analogy (due to Gamow) from optics will illustrate the
kind of phenomenon we are dealing with here. If a beam of light is
incident on the boundary between two media with an angle of inci-
dence greater than the critical angle, then, according to the concepts
of geometrical optics, we will have a total reflection of the incident
beam — the presumption being that all the light will be reflected
at the surface separating the two media and that no disturbance in
the second medium occurs. However, when this same problem is
treated by the methods of the wave theory of light, it is found
that there is, in fact, a finite disturbance in the second medium as
well, which is appreciable for a distance of the order of a few wave
lengths of light and falls off exponentially as we go farther out in the
second medium. This disturbance in the second medium (which is
predicted on the wave theory of light, and verified by experiment)
for angles of incidence greater than the critical angle has no inter-
pretation in the language of geometrical optics. In exactly the
same manner the passage from classical mechanics to quantum
mechanics allows the possibility of particles penetrating potential
barriers, a feature which would be impossible to interpret in the
language of classical mechanics.
For practical purposes it is sufficient to consider for the potential
energy between two particles of charges Z,e and Z.^e the form
V'(r) =
r
A\
(17)
II
(r < r*} .
(18)
and
STELLAR ENERGY 461
V{r), considered as a function of r, is shown in Figure 36. The
potential energy is maximum at r = r*, where it has the value
. (19)
Vb is sometimes called the top of the potential barrier. The energy
of the a-particles = 2) emitted by radioactive bodies is much less
than the appropriate Vb- As we have already indicated, quantum
Kj(;. 3f). 'Fhc nuclear niodcl for the ciilculaLiun of the transparency factor
mechanics allows us to calculate the probability of an a-particle
initially inside r = f * being emitted. We can then evaluate the life-
time of the radioactive nucleus. The calculation is a straightforward
piece of analysis in quantum mechanics, and we shall not go into
the details of the derivation (reference may be made to the litera-
ture quoted in the Bibliographical Notes at the end of the chapter).
The result of such calculations is to give, for the number of particles
463
STUDY OF STELLAR STRUCTURE
emitted per second by a radioactive nucleus of charge Z^e (assumed
to be at rest), the expression
X' = Va
- eY" ’
(20)
where M2 is the mass and E the energy of the emitted particle- -
in the case of a-decay the emitted particle is, of course, an a-particle
— and where the Gamow exponent, G, is given by
G =
h
(21)
In (21) the upper bound of the integral, r^, is defined in such a way
that the integrand vanishes at r = The integration of (21) is
straightforward, and the result can be expressed in the form
^ {^2M ZiZzt'' f ^
G = 9?777- ,
h
(22)
where
K
^ = rr > == cos ^ a:'/-* ~ — .v)'/- . (2^)
From (22) wc see that G increases with increasing nuclear charge,
decreasing energy of the emitted particle, and decreasing nuclear radius
as defined by r*. That this should be so is intuitively obvious, since
increasing Zi and decreasing E and r* imply that the particle will
have to penetrate a higher and a broader potential barrier, which is
naturally more difficult. Equation (20) can be interpreted by the
statement that the “half-life”*' is given by
log 2
r
Equation (24) can be alternatively expressed as
(24)
To
W
(25)
The “half-life” is defined as the time during which half the number of particles are
emitted.
STELLAR ENERGY
463
where JV, the transparency of the potential barrier, is given by
W = (26)
and To is the lifetime without the potential barrier (tq 3.3 X 10“^^
sec. for radioactive nuclei) ; will be, classically speaking, the num-
ber of times the a-particlc will hit the inner wall of the potential hole
per unit time. Equation (25) can, therefore, be interpreted as fol-
lows: W is the probability, per collision, for a particle to penetrate
the potential barrier.
The expression for the transparency of the potential barrier which
we have just given is, in fact, quite general and is precisely what
is needed to calculate the probability of a particle penetrating in-
to the potential hole from outside — more precisely, W is the proba-
bility, per collision, that a particle of charge ZaC and energy E will
penetrate into another nucleus of charge Z,c, when the latter nucleus
is assumed to be at rest. The modification of the transparency fac-
tor, IV y when both the particles are in motion is obvious; we then re-
gard E as the total kinetic energy of the two nuclei in a system of
reference in which the center of mass of the two particles is at rest;
further M is to be regarded as the reduced mass of the system,
M =
MM.
M, + A/a ’
(27)
where Mi and M. arc the masses of the two nuclei considered.
I’he Gamow exponent can therefore l^e written as
G =
A/. A/a ,
Ml + Mj hE^f^
(28)
Imr most stellar applications a simplification of W is possible. As an
empirical fact, it is found that
^ o. yoZiZa.Ip*/’^ million electron volts . (29)
If we remember that a million electron volts corresponds to a
temiie rat lire of the order of io‘^ degrees, it is clear that we can ap-
j)r()ximale /'(j), given by (23), by its value for x — > o. Since
TT
2
O ,
as
.V
(30)
464 STUDY OF STELLAR STRUCTURE
we have
^ VM^ + M^) kE'/^ ■
(,U)
Finally, the transparency is given by
W = (,^j)
4. The penetration of nuclear barriers by charged par tides with
thermal velocities , — ^We shall now calculate the number of successful
captures of a nucleus of charge by another nucleus of charge Z,r
which occur in a system at temperature T and in which the distribu-
tion of the velocities of the different particles is given by Maxwell’s
law (chap. x). It should be mentioned here that each successful cap-
ture does not necessarily imply a transmutation- it is possible that
the captured particle may be re-emitted. We shall return to this
question in § 5.
Now, the number of collisions (per unit volume and per unit time)
in which the total kinetic energy of the relative motion of the collid-
ing particles lies in the range E and E + dEis given by
2NiN2(r\2
{kTyt^
[2r(M^ + M,)
MM,
L
■ f/
where tto-J, is the effective cross-section for the collisions and .V , and
Ni are the numbers of nuclei per unit volume of the two sorts in the
system.
An approximate expression for o-.j can be given. If we represent
the colliding particles by plane Dc Broglie waves, then for “hea<i-
on” collisions the collision cross-section is approximately given by
the square of the De Broglie wave length which characti-rizes the
incident particles in a system of reference in which the center of
mass of the two particles is at rest. Thus,
irh^ _
~ 2ME
i.iO
where M is the reduced mass (cf. Eq. [27]), v is the relative velocity
between the two particles, and E has the same meaning as in eciua-
■> R. H. Fowler, Statistical Mechanics (2cl ed,, Cambridge), p, O65, I^:q. (18(15).
STELLAR ENERGY
tion (33). If we consider a non-head-on collision which is character-
ized by a relative angular momentum, then to obtain the appropri-
ate cross-section we must multiply (33') by a factor which is very
small compared to unity. Hence, it is sufficient to restrict ourselves
to head-on collisions only.
Now the probability that a collision will result in the capture (and
thus lead to the possibility of a transmutation) is given by the fac-
tor W. Hence, according to (33) and (33'), the total number of pene-
trations occurring in the system per unit time is given by
N,N,h^
(34)
II
(35)
0 = {
\ M, + mJ hikT)''^ f •
(36)
Equation (34) can then be written as
r MM, +
2 T{kTy'^ I M,M, J Jo ®
I'o evaluate (37) we note that the exponential term in the integrand
of the expression has a sharp maximum at
— I -f- Q^y = 0 or y =
Since the exponential term falls off very steeply on either side of
we write
y = « + (?* (39)
and regard u as small, since the contribution to the integral arises
essentially from the immediate neighborhood of y = Q®. We then
find
y -t- 20’r'^' = ^ g! + 0(«.’) . (40)
466
STUDY OF STELLAR STRUCTURE
We can therefore write for the integral in (37)
/*oo
3
g-aQ* I e .
(40
The quantity is generally quite large compared to unity, and we
can write with sufficient accuracy
26
= 2 yjj Qe-^0‘ . ( 42 )
Hence, the number of encounters which result in successful captures
that occur in the system per unit volume and per unit time is
given by
(4.0
(44)
(44')
Substituting for Q according to (36), wc have
N N ( M, +
3 '''’ {kry^ \ MM, ) ''
where we may notice that
= , (2 (irZ,Z,r^Y-^ i
n m, + mJ \~2ir) lVfy>-
Inserting the numerical values of the atomic and other constants iu
(^) and (44O, we find that we can express the number of penetra -
tions, P{Z,; Z.), per gram of each of the two sorts of nuclei in the
following form :
Log P(Z,; Z,) = 39.480 + Log (~' 'm {ZiZ^''> |
_/ A xA^Z^Z.X yf'^ 1 , 850 X ro-^ 2 L()<r 7’
\ /li + A 2 ) 7’«Ai , »
where A, and A, are the atomic weights of the nucl(‘i of ehargi's
1 otinTfir^''- <-<>ncerninK
urn rife. ^ transmutations occurring at none(,uilil,ri-
um rates are the fundamental physical processes, we have for the
(4.=; '
STELLAR ENERGY 467
contribution to the rate of generation of energy by the transmuta-
tions of the kind we have been considering
e(Z.Z.) = constant , (46)
where Xi and are the abundances with which the nuclei of the
two sorts occur. Before discussing the important formula (46), we
shall first make a few remarks on its derivation. We assumed that
the distribution of the velocities of the nuclei involved is according
to Maxwell’s law. As a criticism of this, it may be argued that
transmutations generally lead to the emission of particles of very
TABLE 31
T (in Millions of Log P (t; 2)
Degrees) (SeeEq [45))
1 7.8s
2 13-27
3 iS-Sg
4 17-54
5 18.70
10 21.78
20 24.20
30 25.35
40 26.07
considerably higher energies than, would correspond to the tempera-
ture T. But this fact is not of great importance. I’hesc high energy
particles which arc emitted during some types of transmutations con-
sidered will be rapidly slowed down because of the very eflicient
stopping power of the stellar material. Thus, so long as we are not
concerned with physical processes which arc or lo"-*’ times less
frequent than those which are clue to the particles with thermal
energies, it is safe to assume a Maxwellian distribution of velocities
for the nuclei taking part in the capture processes.
Returning to (44), we sec that, according to this formula, the
penetration of protons into the lighter nuclei is easily possible under
the conditions which we have derived for the stellar interiors.
Table 31 illustrates the point.
Another very important feature now becomes apparent. Trans-
mutations by the capture of protons can occur only with elements of
very low atomic number. Because of the occurrence of in
468
STUDY OF STELLAR STRUCTURE
the exponent in (46), the captxire even of protons by the heavier
nuclei becomes extremely unlikely.
If we turn to a different aspect of the situation, it may be argued
that the rate at which captures occur for T < 4 X 10^ is exceedingly
slow. But this question can only be settled by actual integrations
for stellar models with an underlying law for the energy generation
of the type (46). Integrations for one such set of configurations has
been made by Steensholt, who finds that the process considered
here is quite sufldcient as a source of stellar energy. Finally, atten-
tion may be drawn to the extreme sensitiveness to temperature of
the law (46) ; it is this circumstance which led to the belief,® to which
we have referred at the end of § 2, that models built on the law (46)
are likely to be unstable.
5. Von W eizsUcker' s theory. — 'SSft have seen that the penetration
of protons through the potential barriers of the lighter nuclei occurs
in stellar interiors and that they will also suffice — with an adequate
supply of the lighter elements — as a source of stellar energy. As a
typical example we may consider the capture of protons by the
lithium nucleus. Now this capture does not lead to the formation of
a nucleus of a higher atomic number — instead, we have a disintegra-
tion process:
2%He. UlY
We shall consider presently other examples of captures which result
in similar disintegration processes, but it follows that we shall have
an increasing proportion of the lighter nuclei. It is thus clear quite
at the outset that we have to distinguish carefully between syn-
thesis processes and disintegration processes — the German words
Aufbauprozesse and Abbauprozesse, which von Weizsacker has intro-
duced, are very much more expressive. At this stage we should con-
sider three possibilities:
a) That the heavy elements like lead, thorium, and uranium arc
now continually being formed in the stellar interiors and that all
s Not confirmed by Cowlings investigation 94, 7^8, 1934) on the stability
of such models.
® In writing equations representing nuclear reactions, we shall adopt the convention
of prefixing the letter denoting the element on its upper left-hand comer by its atomic
weight and on its lower left-hand comer by its atomic number.
STELLAR ENERGY 469
the heavier elements now present have been synthesized in the
stars during the past.
b) That a great (or an appreciable) fraction of the heavy elements
now present in the stars have been formed at some earlier stage, and
that at the present time, though we have a further synthesis of these
elements, they do not occur at a sufficient rate (or have occurred for
a sufficient length of time) to account for the actual abundances of
the different elements.
c) That all the heavy elements now present in the stars have been
formed at some earlier stage, and that at the present time we have
only a further transformation of hydrogen (involving, principally,
proton captures) into the lighter elements.
These three possibilities cannot of course be sharply distinguished.
The difference between them is mainly one of degree, and we can
easily conceive of a variety of other “intermediate” possibilities.
However, as a working hypothesis, the second and the third have
the disadvantage of not being capable of being made quite definite
at present; in any case, need for the other possibilities can be felt
only by attempting to follow the full consequences of the first
hypothesis. This is the procedure von Weizsacker has followed.
The fundamental assumption, then, is the following:
Apart from secondary effects of minor importance, the transmutation
of elements is the entire cause of the presence of all elements in the stars;
they are all being synthesized continually in the stars which are assumed
to have started as pure masses of hydrogen; further, transmutations are
the only source of stellar energy.
The foregoing hypothesis, which we shall refer to as the “von Weiz-
sacker hypothesis,” is mad^, it will be understood, entirely for the
purpose of having a definite working basis, the partial failure or the
complete success of which will indicate the necessity or otherwise of
considering the other possibilities which we have mentioned.
From the von Weizsacker hypothesis we can draw certain immedi-
ate inferences. First, it is clear that the lighter elements are formed
by processes involving proton captures. These processes will be the
most important among those in which the transmutations are caused
by the capture of charged particles because, as we have seen in
chapter vii, hydrogen is abundant in stellar interiors, and also be-
470
STUDY OF STELLAR STRUCTURE
cause the occurrence of {exp — {Z\Z\IT^')\ in the formula giving
the number of penetrations makes the capture even of a-particles
very much less probable than the capture of protons. Second, the
occurrence of {Z\Z\/TY^^ in the exponential in (46) shows that even
proton captures cannot be of any significance in the synthesis of
the heavier nuclei. Von Weizsacker’s hypothesis therefore requires
that some other physical process is fundamental for the synthesis
of the heavier elements. Now the experiments of Fermi and others
have shown that neutrons can be captured by the heaviest nuclei, so
that it is plausible that this is the physical process responsible for
the synthesis of the heavier elements in stellar interiors. We cannot,
however, assume that there are free neutrons present in stellar in-
teriors. The experiments of Fermi have again shown that the cross-
sections for the capture of neutrons by the atomic nuclei are so large
that, even if there were an appreciable amount of neutrons to start
with, they would all have disappeared in a very short time. Wc
thus infer that the only possibility consistent with von Weizsackcr’s
hypothesis is that there must be a source for a continuous supply of
neutrons, and that the neutrons are formed as a by-product in such
transmutations as do occur under the conditions of the stellar in-
teriors. We shall now consider these questions in greater detail, fol-
lowing von Weizsacker.
(i) Transmutations due to proton captures. — As wc have already
seen, among the transmutations arising from the capture of charged
particles, those due to the capture of protons by the light nuclei are
by far the most important. We shall now consider more closely the
transmutations that can thus occur. In doing so we must distin-
guish between the synthesis and the disintegration processes. Now
an empirically well-established rule which can be used for this pur-
pose is the following.
The capture of a proton by a light nucleus can lead to a synthesis of
a nucleus of a higher atomic number if, and only if, a disintegration
process is not possible from pure energy considerations.
This rule can be understood by the use of the method of descrip-
tion of nuclear phenomena introduced by Bohr: When a proton
penetrates through the potential barrier into an atomic nucleus,
we have first the formation of an intermediate nucleus which in gen-
STELLAR ENERGY
471
eral will be in an excited state; this intermediate nucleus can follow
one of three courses: it re-emits the captured proton, or it emits
some other particle (generally an a-particle),^ or, finally, it drops to
the ground state with the emission of a 7-ray. If the first possibility
occurs, we have a simple scattering phenomenon; if the second, a
disintegration process; and if the third, a synthesis process. Now,
since the lifetime of an excited nucleus with respect to 7-emission is
long compared to the analogous ‘lifetime” To (introduced in § 3,
Eq. [25]), it is clear that (unless the potential barrier is very high and
broad and the energy of excitation of the intermediate nucleus is
distributed among the nuclear constituents) we shall, in most cases,
have the emission of a particle (if it is at all possible) before there
has been time enough for a 7-emission. In the case of the lighter
nuclei the potential barrier can be penetrated without undue diffi-
culty; and since the number of nuclear constituents is small, the
energy of excitation of the nucleus is not distributed quickly among
the other particles. We thus see that for the lighter nuclei we have a
pure energy criterion for distinguishing a disintegration from a
synthesis process. Thus if we compare, for instance, two nuclei, one
of which after the capture of a proton is able (from considerations
of energy) to emit an a-particle and the other not, then in both
cases a synthesis is a priori improbable. But while in the former case
we almost always have a disintegration, in the latter case we will
have the re-emission of the proton. In the second case, since the
re-emission of the proton does not produce any elTective change, it is
clear that the occasional synthesis which can occur is the only one
that matters. Thus, while in the first case every successful penetra-
tion of the nucleus by a proton will be quite invariably followed by
a disintegration, in the second case we have a synthesis only in a
small fraction of the total number of successful penetrations. Thus,
in the latter case we shall have to multiply the expression (44) for
the number of penetrations by another factor which gives the
probability of a synthesis occurring.
Now the probability for the occurrence of a transmutation with
the emission of a 7-ray after the successful penetration of a particle
is given by the ratio of the probability of the emission of a 7-ray
^ If this is possible from cner)j;y considerations.
472
STUDY OF STELLAR STRUCTURE
^ sec"^ (for the order of energies involved) and the proper
frequency hlMr*^ sec-") of the particle oscillating inside
the nucleus. Hence, the factor by which we have to multiply (44)
in order that we may obtain the number of transmutations is given
by
M^M,
M, + M,
(48)
The total number of transmutations occurring in unit volume and
in unit time is therefore given by®
24 / 3^/3
31/a {kTY^
(49)
Having thus settled as to when a synthesis (as distinguished from
a disintegration) can occur, we shall next consider the stability of
the nuclei synthesized by proton captures. We shall now have to dis-
tinguish between stable nuclei and those which are jS active; jS-decay
of the unstable nucleus can consist either in the emission of electrons
(i 3 " -decay) or positrons (jS^-decay). Remembering that, according
to current views, the nuclear constituents are protons and neutrons,
we shall have Z-protons and {A-Z) neutrons in a nucleus of charge
Z and mass A. We shall now state the following rule:
Stable nuclei are those in which the number of protons in the
nucleus is equal to, or i less than the number of neutrons, according
as the mass-number A is even or odd. All, other nuclei are unstable;
nuclei with an excess of protons being active and those with an
excess of neutrons being active.
When some stable nuclei capture protons, they emit a-particles.
Thus,
ILi - 1 - 1/7 2 ille , ]
+ (50)
* See a paper by G. Gamow and E. Teller {Phys. Rev., 53* 608, 1938) that has since
appeared. Gamow and Teller have in addition considered “resonance penetrations”
and indicate the importance of the consideration of such processes.
STELLAR ENERGY
473
In the boron-proton reaction the final nucleus \Be (which is general-
ly left in an excited state) disintegrates almost immediately into two
a-particles.^ It will be noted that in all the foregoing cases we have
the formation of especially stable nuclei {\He, ”C), in which the
number of neutrons and protons are equal. It will be found that by
proton captures we cannot have a further synthesis of elements
“over” these nuclei. On the other hand, if the capture of a proton by
a stable nucleus leads to an intermediate nucleus which, according to
our energy criterion, cannot emit an a-particle, then we will have the
synthesis of a nucleus of a higher atomic number which will be jS'*"
active. We shall, however, see in (2) below that these i 3 + active
nuclei can be “stepping-stones” for the synthesis of still higher mem-
bers of the periodic table.
It follows from what we have said that by successive captures of
protons we cannot (under stellar conditions) have the synthesis of
nuclei heavier than, say, oxygen. This is so for two reasons, both of
which work in the same direction. First, by successive proton
captures wc arc led (according to our stability criterion) to the syn-
thesis of nuclei which will be more and more active. Second, the
increasing nuclear charge will decrease exponentially the probability
of a proton penetrating the nucleus (because of the factor {exp —
constant [Z\Z\/TY^^\ in [44]). Since, for a further synthesis over
a jS'*' active nucleus, the proton will have to be captured before the
jS^-decay, it is clear that wc shall soon come to a stage where the life-
time of the nucleus for |8+- decay becomes less than the mean inter-
val of time between two successive proton penetrations. This con-
dition clearly sets an upper limit to the atomic number of the ele-
ments beyond which a further synthesis by proton captures can-
not be possible. The actual point in the periodic table where further
synthesis by proton captures in elTect ceases will be, however, very
much earlier, since a successful penetration does not (as we have
seen) imply a successful synthesis.
Summarizing, we can say that nuclear reactions involving proton
captures result essentially in an accumulation of the lighter nuclei
which (as wc shall sec in greater detail below) in turn act as
catalysts in the production of further a-particles. This, then, is the
'* See N. Feather, Nuclear Physics (Cambriclj^c, En ;4 in'l, rQ36), p. igi.
474
STUDY OF STELLAR STRUCTURE
fundamental physical process which is effective as the source of
stellar energy.
(2) Nuclear transmutations by proton captures as an autocatalytic
chain of cyclical reactions . — We shall now examine more closely the
actual nuclear reactions involving proton captures. Because of the
incomplete nature of our information concerning the masses of
some of the lighter nuclei, the following discussion (due to von
Weizsacker) should be regarded as partly tentative. However, the
discussion discloses certain characteristic features which are likely
to survive in the future discussions concerning stellar energy.
The natural starting-point is clearly the consideration of nuclear
reactions in which both protons and a-particles are involved. At
this point we encounter a difficulty; laboratory investigations have
so far failed to disclose the existence of a nucleus of mass 5. Von
Weizsacker believed that the existence of \Li and \He could be con-
jectured. However, according to Bethe, the more recent experi-
ments on artificial disintegrations exclude more or less definitely
the possibility of a nucleus of mass 5.^® In spite of this, we shall out-
line the nuclear reactions considered by von Weizsacker as illustra-
tive of the nature of such discussions.
Von Weizsacker considers two possibilities:
ILi is active
(1)
and
ILi is stable .
(11)
Let us first consider case I. The course of the reactions to bo de-
scribed can be followed by referring to Figure 37. The first nuclear
reaction is (/) %IIe + = \Li^ which by hypothesis is active.
We then have either (i. i) a jS^-decay of ^Li, in which case we would
have \Li = {He + 18+, or {1.2) a further capture of a proton by
ILi before it decays; in the latter case we have ^Li + \ II = ^Bc, and
this nucleus (according to our stability criterion) must be strongly
active. In case (i.i) the most probable reaction is (r.ii)
llle -h \II = llle -b ID. Wc sec that we have now completed a
cycle. The a-particlc with which wc started has been recovered, and
the whole cycle- (/), (i.i), (i.ii ) — can now be repeated; the net
“ See, however, reference 20 in the Bibliographical Notes (p. 486).
Fig. 37. — The nuclear reactions involving proton captures as autocatalytic chains
476
STUDY OF STELLAR STRUCTURE
result of each cycle is that a deuteron and a positron” have replaced
two protons. The helium thus acts as a catalyst in this cyclical
chain of reactions. We shall postpone consideration of the part
which ID plays in further reactions and return to the case (r.s).
Here we again have two possibilities: either {1.21) the |8+-decay of
^Be takes place before a proton capture (resulting in the formation
of the stable isotope ^Li of lithium) or (1.22) a proton is captured
by iBe before the |8+-decay (in which case we have the synthesis of
IB, which must be strongly /S'*’ active). In case (1.21) the further
capture of a proton by ^Li will result in {1.211) \Li + \E =
iHe + \Ee. Again a cycle has been completed; the a-particle with
which the chain of reactions — (r), (1.2), (r.21), (j.2ij)— started has
been recovered, and the three protons which took part in the re-
action chain have been replaced by \Ee and a positron. We shall
consider the further reactions in which lEe is involved a little later;
but, returning to case (1.22), we can assume that \B is so strongly
|8+ active that we effectively have only one possibility, namely,
{1.221) (the /3+-decay of IB to IBe, which must also be active).
With IBe we again have the two possibilities: {1.2211) the /3'''-decay
of IBe to the stable ILi, and {1.2212) the capture of a proton by \Be
(before |8+-decay) to form the /S '" active \B. In the case {1.221 1) the
further capture of a proton will result in {1.22111), the disintegra-
tion of ILi into two a-particles. At this point another cycle has
been completed. We note, however, that at the end of this cycle
we have two a-particles for every one with which the cycle — (/),
{1.2), {1.22), {1.221), {1.2211), (j.22i7/)— started. In other words,
the cycle which we have just considered can be called (in the lan-
guage of chemistry) an “autocatalytic cycle,” since the net result
of the cycle is to increase the amount of the catalytic agent {\Ee)
present. In the case {1.2212) the iS^-decay of will lead to the
stable (or weakly a active) \Be. The latter, on capturing a proton,
will form the jS"*" active \B, which after its jS^-decay will result in
the formation of the stable \Be. Finally, the most probable reac-
tion, which would result on \Be capturing a proton, will be its dis-
integration into two a-particles and a deuteron. Another auto-
“ The positron would later combine with an electron and emit two 7-rays.
STELLAR ENERGY
477
catalytic cycle has ended. It is important to notice that again we
have the formation of deuterons.
We have yet to consider the further reactions in which {He and
\D are involved; but before doing so, we may note that if we con-
sider case II (where ILi is assumed to be stable), the whole se-
quence of the reaction chains is exactly the same as in case I, except
that the first cycle — (i), (i.i), {i.ii ) — does not exist for case II.
This results in one important difference between the two cases;
if \Li is jS'*' active, we have the formation of deuterons at an early
stage in the reaction chains, whereas they appear at a relatively later
stage in the sequence of the reaction chains if ILi is a stable nucleus.
We have considered the reaction chains among the lighter nuclei
up to the synthesis of a stable isotope of beryllium mainly for the
reason that, if ILi should be stable, then precisely at this point do we
have the formation of deuterons; as we shall presently see, the
production of deuterons is important in the further development of
the von Weizslicker theory. There is, however, not much point in
continuing the reaction chains to include formally the synthesis of
the higher members of the periodic table, as we have already ex-
plained (toward the end of [i], above).
We shall now consider the reactions in which yfle and \D take
part. If lllc is active it would be transformed to It can also
capture a proton and synthesize what must certainly be strongly
active, namely, \Li (which, after its jS'^-decay, will result in the
formation of illc).
As for the deuterons, since the probability of the capture process,
+ (A)
(with \Ile following the chain of reactions already described), is
probably very small, we must also consider deuteron-deutcron re-
actions:
fZ? + = \IIc + In
(BI)
fZ) + fz; = jr + \II .
(BII)
The reactions BI and BII are the most efficient artificial transmuta-
tions that have so far been elTccted in the laboratory; we can esti-
478
STUDY OF STELLAR STRUCTURE
mate the capture cross-section for BI and BII to be about lo® times
the capture cross-section for A. It is probable that, except in the
earliest stages in the history of a star, deuterons are more than io“^
times as abundant as protons, so that the reactions BI and BII
become more important than A. We thus see that in the reaction
cycles going on among the lighter nuclei we have found a process
which will serve as a source of neutrons, the need for which we have
already explained in § 5.
We thus see that the characteristic features of the nuclear reac-
tions we have considered are (i) the nuclear reactions go in cycles;
(2) in the cycles the a-particles play the part of catalytic agents,
some cycles being even autocatalytic; (3) the reaction chains lead
at some point to the production of deuterons; (4) the deuterons, if
more than lo^^ times as abundant as hydrogen, will serve as a source
of neutrons. These are the essential points in von Wcizsilcker’s
discussion.
In order to simplify our discussion we shall, following von Weiz-
sacker, consider the following model reaction chain :
;// -f ^Jle = ILi ,
JIf = +
and
\D + \D = \Ile + In ; \D + \D = ]T -f ;// .
(50
(52)
llle and \T again lead to the formation of a-particles.
(3) Synthesis of the heavy elements by neutron capture ~-Ov\r dis-
cussion in (2) has brought out clearly the inadequacy of transmuta-
tions involving proton captures for the purpose of synthesizing the
heavy nuclei. As we have seen, we must look for the synthesis of
the heavier nuclei in transmutations involving neutron captures; for
this purpose we need a continuous supply of neutrons. We have
already shown that the reaction chains among the lighter nuclei do,
in fact, include nuclear reactions (if the deuterons are more than
io“s as abundant as hydrogen) which will serve as a source of neu-
trons. But before we can be sure that the neutrons do synthesize
the heavy nuclei, we should make certain that the neutrons are, so
STELLAR ENERGY
479
to say, not wasted by recombining with protons to form deuterons.
The deuterons thus formed may again produce neutrons, but it is
clear that for each such cycle there will be a reduction of the neu-
trons by a factor of 4. However, there is no very great danger of
this happening, for the capture of neutrons by atomic nuclei takes
place by what is called a ^'quantum mechanical resonance”; if the
incident neutron has an energy nearly equal to an energy level for
the neutron inside the nucleus, then we have a kind of ‘‘resonance”
which makes the transition probability for the neutron penetrating
the nucleus especially large; further, the energy levels for the neu-
trons inside the heavier nuclei lie very close together — it is this last
circumstance which makes the capture of neutrons by the heavy-
nuclei relatively easy. However, the energy levels of the neutron
inside a \D nucleus rapidly become spaced farther and farther apart;
and (according to von Wcizsacker) in the energy range for the
neutrons corresponding to thermal energies in stellar interiors there
are very few resonance levels. This would make the capture cross-
section for the neutron-proton reaction very small compared to what
it is for the heavier nuclei. It thus seems safe to conclude that the
neutrons produced in the deutcron-deuteron reaction will be avail-
able for the synthesis of the heavier elements.
There is one further point which should be mentioned, namely,
that the liberated neutrons will soon attain thermal energies cor-
responding to the temperatures in the stellar interiors. The phenome-
non we encounter here is essentially the same as that which has
been found in laboratory experiments in which a block of parafhn
slows down high-energy neutrons so quickly that they very soon
attain the thermal energies corresponding to the room temperature;
also stellar material with its abundance of hydrogen is, with respect
to its stopping power of the neutrons, not very different from a
paraffin block. To avoid misunderstanding, it should be stated that
there is no contradiction here with our earlier remarks that the
capture cross-section for the neutron-proton reaction is small com-
pared to that for reactions with the other elements; for the probabil-
ity of elastic scattering is more than one hundred times the cor-
responding probability for the synthesis of a deuteroii.
The synthesis of the heavy elements being thus made plausible,
48o
STUDY OF STELLAR STRUCTURE
we may note the physical factors which would govern such trans-
mutations: (i) the rate at which the neutrons are hberated, and (2)
the density of the resonance levels for the nucleus concerned in the
energy range of the neutrons corresponding to the temperatures in
stellar interiors.
IQA
lUT
133
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I32(
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t3i
(
^.RqC, •^;20m ( )
I30<
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I29(
)AeB;(9;36rn<
)rThCj6t;p;lh
I28(
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)AcC;p6;2m a ’
127
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82
83
84
• Stable nucleus o Radioactive nucleus
I'lG 38 The synthesis of the heavy elements over lead (Z=H 2 )
We now come to what is perhaps a difficulty in the von Weiz-
sacker theory in its present form. If we assume that all the heavy
elements are synthesized at present in stellar interiors, we shall see
that the rate at which neutrons should be captured by the heavier
nuclei must be extremely rapid. Consider, for instance, the syn-
thesis of the elements above lead and bismuth (see Fig. 38). In
this region of the periodic table we have the elements of atomic
STELLAR ENERGY
481
number 84 (the C'-products) which are extremely a active and
which on a-disintegration result in the isotopes of lead. These C'-
products have half-lives with respect to a-decay which are extremely
short (the half -lives are noted in Fig. 38). Iri order, then, to synthe-
size Th and it is clear that we must have neutron captures in
such quick succession that we can go forward in the synthesis
processes in spite of the decay which is taking place all the time. The
synthesis will first continue along the sequence of the isotopes of
bismuth (Z = 83, C-products). The most difficult ‘‘barrier” here is
RaC^ which is jS active with a half-life of 20 minutes — RaC on jS-
decay goes over into RaC^ which in 10“^ sec will go over into RaD
(an isotope of lead) on a-disintegration. Further, the isotopes of
bismuth with masses greater than RaC (i.e., greater than 214) must
be all /S active with much shorter lives than RaC, It is thus clear
that RaC should capture four successive neutrons (all consecutively)
before any jS-decay occurs, in order that the jS active isotope of
bismuth with mass 218 may, on its jS-decay, result in the synthesis
of RaA , the lifetime of which for a-dccay can be measured at least
in minutes. We thus see that for the synthesis to go forward from
lead it is necessary that the neutrons be captured at a very rapid
rate; also, we cannot allow the mean interval between successive
neutron captures to be less than, say, i minute. It is only then that
we can have for instance the synthesis of uranium. Von Wcizsacker
points out that this need for a very rapid succession of neutron
captures cannot be an exception, in so far as uranium and thorium
are not much less abundant than lead. It should be mentioned,
however, that for the synthesis of the “moderately” heavy elements
we should very probably not need so rapid a reaction-rate.
There are various problems suggested by the conclusions reached
above. We shall consider some of them in (4) below, but we may
note here that precisely the extreme a-activity of C'-products will al-
low a more or less straightforward explanation of the relatively large
abundance of lead. Indeed, proceeding on similar lines, von Weiz-
sacker has examined the abundance of the different elements from
the present point of view in some detail; he believes the possibility
of the present scheme being sufficient (at least as a working basis)
STUDY OF STELLAR STRUCTURE
482
for a complete understanding of the general (Z, abundance) rela-
tion.” We shall not, however, go into these matters further here.
(4) Astronomical implications— So far we have been concerned
with general ideas j it now remains to examine the astronomical
implications of the von Weizsacker theory which we have described.
There are a great many details that remain to be worked out, but
we shall consider only two definite consequences of the theory.
i) The helium content of the stars— li we consider the model
process (51) and (52), we see that the mass of helium that is formed
at the end of each cycle is much greater than the mass of the neu-
trons liberated. We can formally combine equations (51) and (52)
and write
2\H + iHe = iHe-^\D, (5.O
\D = mn + lUe -h \T + \U\ , (54)
and
mEe + lT+ 2 \H] ( 55 )
Combining the foregoing, we have
I proton — > ^ a-particle + J neutron . ( 50)
Thus, as the net result of a complete autocatalylic cycle corrc’spoinh
ing to the model process, we have the liberation of one neutron for
every two a-particles synthesized; in other words, tlu‘ mass of
helium synthesized is eight times the mass of the niaitrons liberal t‘<l.
Now according to our discussion in (,0 above, it is the neutrons
which synthesize the heavier nuclei; we can therefore say (luiti*
roughly that the mass of all the elements (other than lu'lium)
synthesized will be equal to the mass of the neutrons libi'raled. W’v
thus see that an immediate consequence of the theory is that tin*
stars should contain a relatively high abim(lanct‘ of helium; in-
deed, according to the theory, we can ex])ect h(‘lium to be as much
as (at least) eight times as abundant as the “metals.” 1 ’his is an
astronomical prediction which should be ca])abk‘ of verification. In
'^Von Weizsacker has since abandoned this hope. Sit ibc Uil.li.)Lrr;iphi< ;d Noirs
(ref. 19) at the end of the chapter.
STELLAR ENERGY
483
chapter vii, § 9, we have already referred to Stromgren’s investiga-
tion of the helium content of the stars from the point of view of the
mass-luminosity-radius relation. From the discussion (see especial-
ly Table 16), Stromgren concludes that the hydrogen-helium-
Russell mixture hypothesis is ‘‘compatible with the observed
masses, luminosities, and radii of the stars.’* This, then, can be re-
garded as supporting von Weizsacker’s theory in a general way.
ii) The role of convection currents . — ^According to the transmuta-
tion hypothesis of the origin of stellar energy, the rate of generation
of energy will be given by a law of the type (46). Now a law of this
type implies such a sensitive dependence of e on T that we should
expect the point-source model to be a suitable idealization for de-
scribing the structure of stars.
Since the nuclear transmutations among the lighter nuclei form
an autocatalytic chain of reactions in which each cycle results in
increasing the abundance of the lighter elements in some definite
ratio, it is clear that the abundance of the lighter elements should
increase exponentially with the number of cycles. If we consider
the sun, the protons which it would contain if it were all hydro-
gen would all be used up after about 200 cycles, which then is the
upper limit to the number of cycles that could have occurred in the
sun. Now in each cycle the neutrons liberated have a mass of J (see
Eq. [56]), so that an average nucleus cannot have captured more
than 50 neutrons. On the other hand, we have seen in (3), above,
that for the synthesis of the heavy elements the neutrons should be
captured at a very rapid rate- in fact, about once a minute. It is
thus clear that the effective mass of the star which can, at a given
instant of time, take part in the nuclear reactions must be extremely
small. We have, therefore, to postulate that there exist convection
currents and that, further, they succeed in effecting a continuous
interchange of matter between those regions in which the nuclear re-
actions arc taking place and the other parts of the star. There is,
of course, no difficulty in admitting convection currents — indeed,
from our discussion in chapter vi, § 2, it follows that in the point-
source model the radiative gradient must necessarily become un-
stable in the central regions. But if there is convection, then, we
should expect the currents to produce a more or less uniform condi-
484
STUDY OF STELLAR STRUCTURE
tion over an appreciable volume in the neighborhood of the center,
and it is difl&cult to see how we can succeed in confining the regions in
which the nuclear reactions are supposed to take place to an ex-
tremely limited volume which the von Weizsacker theory in its
present form requires. It will have to be borne in mind in this con-
nection, that we do not yet have a satisfactory method of dealing
with problems involving convection currents. It is not suggested
that these difficulties cannot be overcome, but they are problems
for future investigations. This is perhaps an unsatisfactory state in
which to leave the subject, but the object of devoting this amount of
space to the von Weizsacker theory is not because it is, as yet, a fully
developed theory but because it introduces some ideas which are
of general importance. There is still a variety of other problems
(e.g., stability) that can be discussed in this connection, but they
are all beyond the scope of this monograph.
BIBLIOGRAPHICAL NOTES
I. H. von Helmholtz announced his estimate of the time scale on the contrac-
tion hypothesis and suggested the meteoric theory of the origin of solar radiation
at a popular lecture delivered at Konigsberg on the occasion of the Kant com-
memoration in February, 1854. But Kelvin showed that “neither the meteoric
theory of solar heat nor any other natural theory can account for the solar radia-
tion continuing at anything like the present rate for many hundred millions of
years.”
Lord Kelvin’s contributions are:
1. Lord Kelvin, Brit. Assoc. Rcpls., 1S61, Part 11 , pp. 27 28 (reprinted in
his Collected Papers^ 5, 141-144).
2. Lord Kelvin, Les Mondcs, 3, 473-480, 1863.
In a popular lecture delivered in 1897 Kelvin gives a most attractive account
of his ideas on the time scale. See —
3. Lord Kelvin, Collected Papers, s, 205-230.
II. For general information on nuclear physics the following references may
be given:
1. N. Feather, N^lclear Physics, Cambridge, 1936.
2. F. Rasetti, Elements of Nuclear Physics, New York: Prentice Hall, T036.
3. G. Gamow, Structure of Atomic Nuclei and Nuclear Transformations, Ox-
ford, 1937.
4. C. F. VON Weizsacker, Die Atomkernc, Leipzig, 1937.
5. H. A. BETHE'and Others, Revs, of Modern Phys., 8, 82, 1936; 9, 69, 245,
1937 -
STELLAR ENERGY 485
Transmutation of elements as the source of stellar energy was suggested by —
6. A. S Eddington, Brit. Assoc, Repls., ig20, p. 45.
7. J. Perrin, Rcd. du mois, 21, 113, 1920.
8. W. D. Harkins and E. D. Wilson, Phil. Mag., 30, 723, 1915.
After the discovery of the theory of a-decay by Gamow and by Condon and
Gurney, the transmutation of elements arising from proton captures was con-
sidered by —
9. R. d’E. Atkinson and F. G Houtermans, Zs.f. Phys., 54, 656, 1929,
Further elaborations of the ideas contained in the paper of Atkinson and
Houtermans are contained in —
10. A. H. Wilson, M.N., 91, 283, 1931.
11. G. Steensholt, Zs.f. Ap., s, 140, 1932.
Further developments are contained in
12. R. d’E Atkinson, Ap. J., 73, 250, 308, 1931.
At the time of Atkinson’s work (ref. 12), nuclear physics was in too rudimen-
tary a stage, and consequently many of Atkinson’s ideas have either been super-
seded or have to be reinterpreted.
1'he general theory described in § 5 is due to —
13. C. F. von Weizsacker, Phys. Zs., 38, 176, 1937.
For a general account of von Weizsacker’s work see —
14. B. Stromgren, Jirg. cxakl. Natunviss., 16, 465, 1935 — particularly pp.
5^9529-
'rhe problem of the helium content is also examined in reference 14.
Hie following investigations have appeared since the writing of the mono-
graph:
15. G Gamow and E. 'Feller, P/iys. Rrv., 53, 608, 1938.
16. G. Gamow, 7 *hys. Rev., 53, 595, 1938.
In references 15 and 16 it is .shown that resonance penetrations of charged
particles can be of great importance if there are low-lying nuclear energy-levels
(with excitation energies of the order of 10 kilovolts).
17. H. Bethe and C. H. Critcii field, Phys. Rev., 54, 248, 1938.
In reference 17 it is shown that the reaction }// 4- }// = ^I) + can occur
at quite appreciable rates under the conditions of the stellar interiors; indeed,
the authors show that this reaction is suflicient to account for the energy genera-
tion in the sun.
18. G. Gamow, Zs f. Ap., 16, 113, 1938. This paper contains an attractive
summary of the present state of the theories of the origin of stellar energy.
During the proof stage of this chapter another paper by von Weizsacker has
been received which carries somewhat further the discussion in reference 13.
IQ. C. F. VON Weizsacker, Phys. Zs., 39, 633, 1938.
In this paper von Weizsacker believes that the dill'iculties mentioned in § 5
(pp. 481 and 483) are so serious that they require the abandoning of the hy-
pothesis made at the beginning of § 5 (p. 469). Von Weizsacker now proceeds
on the basis of the alternative (c) mentioned on page 469.
486 STUDY OF STELLAR STRUCTURE
Also, the possible nonexistence of an atomic nucleus of mass 5 requires the
consideration of other types of nuclear reaction chains. Von Weizsacker (and
also Bethe) now suggest the following chain of reactions in which carbon plays
the role of a catalyst:
Gamow has since come to the conclusion that the foregoing chain of nuclear
reactions represents the fundamental physical processes which serve as the pri-
mary source of stellar energy for the stars on the main series.
A still later paper by Joliot and Zlotowski appears to establish experimen-
tally the existence and stability of IHe,
20. F. Joliot and I. Zlotowski, Jour, d, Phys. g, 403, 1938 (December).
APPENDIX I
PHYSICAL AND ASTRONOMICAL CONSTANTS
TABLE 32
Number
2.9978X10^"
4.801
X io~‘«
9.IOS
Xio-»
I .672
Xio-'-t
1.673
Xio-»-i
6.62
Xio-“T
^ -379
Xio-'is
8.24
Xio'
7-55
X io“*s
9.91
Xio'>
1.231
Xio's
6 .or
Xio”
0.82
Xics
1 .098
Xros
0 .07
X io~*^
1 .985
X 10 - 5 .^
6 .o 5 r
X io"‘
3 • 7‘8o
X 10.^1
+ 4 -f ’3
1 .4109
7.71
Xio*
I . 142
X io-^»
( = 5 -
7.5©)
Velocity of light (cm/sec)
Electronic charge (e.s.u.)
Electronic mass (gm)
Mass of the proton (gm) n
Ma§s of the hydrogen atom
Planck’s constant (erg sec) h
Holtzmann’s constant /.
The gas constant /.///
The Stcfan-Poltzmann constant a =
'Phe un relativistic degenerate constant
1 , 20 W/
I he relativistic degenerate constant
,,,, , 8\jr/ //V.i
I he constants of the equation / A =
of slate of a degenerate gas ] /l/xr' = ^irnh^c^II
'Phe Rydberg constant for inlinilc nuclear mass, , . A’co
'Phe constant of gravitation (dynes cmVgm-*) C
'Phe mass of the sun (gm) 0
The radius of the sun (cm) A’q
'Phe luminosity of the sun (erg/sec) Aq
The absolute bolomctric magnitude of the sun
'Phe mean density of the sun
'Phe unit of length (cm) == ^ '^^-L
MM , Vw B
I he limiting mass (gm)
Logarithm
10.4768
10.6813
£8-9593
24 . 2232
24.223s
27.8209
16.1396
7.9161
15-8779
12.996
15.090
22.779
5 092
5 . 0406
8.8241
33 2078
10.8420
33-5775
0.149s
8.887
34.058
487
APPENDIX II
THE MASSES OF THE LIGHT ATOMS
TABLE 33*
Atom
Massf
Prob-
able
Error
XioS
Stability
Atom
Massf
Prob-
able
Error
XioS
Stability
e
0.0005s
0
12.019
TO
/S'" active
s
'\c
11.01526
35
/3+ active
In , . . .
1.00897
6
^ active
^IC
12.00398
10
stable
IH...
1.00813
2
stable
'IC
13.00761
15
stable
(D) \H. . .
2.01473
2
stable
^iC
14.00767
12
iS“ active
(T) \H...
3.01705
7
stable
^N. . . .
13.01004
13
active
3.01707
12
stable];
»N. . . .
14.00750
8
stable
iHe...
4.00389
7
stable
«iV. . . .
15.00489
20
stable
IBe...
5 0137
40
n\ active
'5 Ar. . . .
16. on
200
/S' active
iHe...
6.0208
SO
jS” active
10
15.0078
40
/S"^ active
6.01686
20
stable
10
16.00000
0
stable
7.01818
i8
stable
^lO
17.00450
7
stable
hi...
8.025
100
i8~ active
10
18.00369
20
stable
. .
8.00792
28
aactive(?)
'W
17.0076
30
jS * * * § ■ active
9 01504
25
stable
1 /-'
18.0056
40
/8^ active
‘“Be. . .
10.01671
30
active
'IP
19.00452
17
stable
“iB....
10.01631
25
stable
V
20.007
250
/3“ active
•iB....
II .01292
17
stable
*The values given in this table are taken from a paper by H. Uethe Mod, Pliys., 9, mm;).
t To obtain the nuclear mas.scs, the massesof the appropriate numbers of electronsshouhl be sublracled
from the values given.
t This nucleus will probably absorb a free electron and go over into •]/J.
§ More recent e.\perimcnt8 by Joliot and Zlotowski .seem to indicate that is stable.
488
APPENDIX III’
THE MASSES, LUMINOSITIES, AND RADII OF THE STARS
DERIVED HYDROGEN CONTENTS; CENTRAL DEN-
SITIES; AND CENTRAL TEMPERATURES
TABLE 34a*
Visual Binaries
Star
Lok M
Lok L
Sun
0.00
0.00
ri Cas A
- .14
—0.09
11
- - 8,5
— 1 . 16
Oi Kri C
- .70
— 1 . 06
a Aur A
+ . f)2
+ 2.08
B
+ .52
+1.90
a CMa A
+ -87
+ 1 ■ 59
a CMi A
-h .17
+o. 7 f)
C UMa A
+ . 4 >
+ i. 4 «
a Cen A
+ .04
+0. 10
B
— .of)
-0.43
$ Boo A
— . of)
—0.32
B
— .12
—0.83
f Her A
— .02
+0.59
- 8 '' 4 ;^S 2 ^
- -45
~ I . d 0
M Her BC
- 1.87
70 Oph A
- .03
-0.3H
B
“ • M
-o.Hf)
Kr. 60 A
— 0.60
-1.77
Lok R
^ot
Log Pc .
1-/8
Log Tc
0.00
—0.09
—0.25
-0.37
+ 1.20
-i-0.82
+0.25
+ 0.23
+0.2S
4 - 0 . 09
— o.of)
— o.ofj
—0. 10
+0.29
— 0. 12
—0. 10
—0.03
—0. if)
—0. 29
O.3O
0.25
[0.34I
1.88
2.00
[1.29J
0.003
.003
[.ooi]
7.29
7.32
[7.25]
•80
•80
• 81
.45
. 8 f>
• 48 l
■ 891
.4BI
. 12
. 20
[ -.sol
1 -421
1 -sol
|o. 19I
— 1 . 10
—0.06
i-So
1-37
1-45
1 .66
2.00]
1.98I
2.07I
0.99
1.79
1.85
1 -98
2 . 22
2.15
0 — .
0000 0 000000 0000
0000 0 "-OOGO*-* Oi-tCAiOi
6.71
6.99
7-39
7.25
734
7.23
[7.25
7.27
[7.21
713
7.09
7.10I
7 .241
7.25I
715I
* In lljcsir Uiblcs L, A/, and K are expressed in Ihe correspemdinK solar units.
t These values were sui)plied by M. StrbniKreii. Those i^iven in f 1 brackets are for stars too dense for
the theory of chap, vii to be applicable with reasonai>le certainty.
* The diita used in chaps, vii, viii, and xi will be found, occasionally, to differ slightly
from the values given in this Appendix. 'Phe values given here correspond to Kuiper’s
final revision of the observational material and are taken from Ap, 7 ., 88, 472, 1938.
For the data for the stars of the Hyades cluster see Table 15, p. 287.
489
490
STUDY OF STELLAR STRUCTURE
TABLE 3U
Spectroscopic Binaries*
Star
Log JIf
Log L
Log R
Castor Cl
—0.201
— 1 .16
—0.18
C.
-0.247
— 1.24
—0.22
jSAur A
+0378
+ 1.83
+0.43
B
+0.370
+1.83
+0.43
fxi Sco AB
+1.094
+ 3-35
+0.73
VPup AB
+1.265
+3-86
+0-83
YCyg AB
+1.238
+ 4 S 1
+0.77
AoCas A
+ 1-634
+ 5-97
-|-t. 3 f)
B
+1.582
+5.58
+ 1-23
29 CMaA
+1.66
+3-84
+ i. 3 ‘
B
+ I-S 3
■fS -39
+ 1.13
* The values of Xq for /9 Aur A and H are (according to SirfImKren)
0.27 and 0.2.'?, respectively. The corresponding values of (i — fi) are 0.022
and 0.023. The central temperature for both stars is about ig million
degrees.
Fur the other stars in this table the theory of chap.s. vi and vii cannot
be applied with reasonable certainty (cf. chap, viii, §5 0 and 7).
TABLE 34c
Trumpler’s Stars
Star
Log M
Log L
Log R
Q*
NGC 2244, 15
1.76:
5 . 40
0 . 66
3
8
I 90
4.69
o.St)
2
NGC 2264, 60
2.18
5 33
0.S2
I
NGC 2362, i
2.47
5 • 73
1 . 28
4
NGC 6871, 2
2.35
5 33
I. 18
()
s
2 .OO
501
1 . 22
5
NGC 7.^80, 1...
1.89
4 89
0 . 96
7
* 'I’he «)rder of reliability of the mca.sured re<l .shifts according to a private comnuinii a-
tion from Dr. Triimpler.
TABLE 34rf
White Dwarks
Star
Log M
Log L
I.og R
Sirius B
— O.OI
-2.52
-I. 71
oa Eri B
“ .35
-2.25
-‘•74
Van Maanen No. 2. . .
+0.53:
- 3 «.‘i
-2.05
APPENDIX IV
TABLES OF THE WHITE-DWARF FUNCTIONS
In the following tables (35-44) the solutions of the differential
equation
JL A.
t (h
(i)
for different values of i/yl and satisfying the boundary conditions
0 = 1 ,
d<t>
dr}
o
()7 = o) , (2)
are given. In addition to the function 0 and its derivative 0', cer-
tain other auxiliary functions are also tabulated. The quantities
p/po, Po/p(v) and — 77^0' describe the physical structure of the com-
pletely degenerate conligurations (see chap, xi, Eq. [83]).
Regarding the accuracy of the tables, it might be stated that
errors exceeding three to four units in the last figures retained are
not expected. The quantities 0 and 0' have been checked by differ-
encing. These tables of the white-dwarf functions (computed by
Chandrasekhar) are published here for the first time.
491
492
STUDY OF STELLAR STRUCTURE
TABLE 35
o,
O. I
O. 3
0.3
0.4
o.S
0.6
0.7
0.8
i.s
1.3
1.4
IS
1.6
2.3.
2.4
2.5.
2 . 6 .
2.7.
2 . 8 .
2.9.
3.0.
3.1.
3-3
3.4
35
3.6
3.8
3-9
4.0
4.1
4.2
4.3
4.4
n
n
4.9.
S-o.
S.i.
5-2.
S 3
0
P/Po
-0'
Po/piv)
I
1
0
1
0
0.998361
0.995041
0.032737
1.00299
0 . 00033
0.993472
0.980348
.064892
I .01197
0.00260
0,985420
0.956463
.095910
I . 02704
0.00863
0.97434s
0.92424s
. 125284
I .04832
0.02005
0.960433
0.884805
.152576
I . 07600
0.03814
0.94391 I
0.839435
-177433
1 . 11032
0.06388
0.925036
0.789525
. 199591
1.15156
0.09780
0.904088
0.736^80
0.681653
.218883
1.20008
0.14009
0.881358
.235231
1.25626
0.19054
0.857140
0.626289
. 248642
1.32055
0.24864
0.831725
0.571479
.259195
1-30347
0.31363
0.805392
0.518140
. 267030
1-47555
0.38452
0.778403
0.751003
0.723410
0.695820
0.467004
0.418618
0.373363
.272331
.275316
.276221
.275294
1- 56739
1 . 66966
1 . 78306
1 . 90833
0.46024
0.53962
0.62150
0.7047s
0.78834
0,331468
0.668404
0.641308
0.293033
.272780
2 . 04629
0,258055
.268918
2.19778
0.87129
0.614657
0.588552
0.226449
.263932
2.36370
0.95280
0.198070
.258032
2 . 54500
1.0321
0.563075
0.172730
.251407
2.74267
X . 1087
0.538289
0,150216
.24A227
.236642
2.95774
1 . 1821
0.514243
0 . 130299
3. 191.^0
1 . 2518
0.490970
0.1 12749
.228782
3.44446
1.3178
0.468492
0. 44682 X
0.097337
0.083844
.220758
.212666
3.7 i 8 .s 8
4.01428
1 3797
1-4376
0.425959
0.072064
. 204582
.190574
4.33338
I 4914
0.405902
0.061804
4.67697
1-5411
0.386640
0.052889
.188692
5-04634
I - 5860
0.368158
0.045157
0.038^63
0.032080
. 180979
S- 44283
1.6288
0.350437
.173467
5.86781
1 . 6670
0.3334S7
. 166x81
6.32268
I . 7017
0.317193
0.027690
. 1591.^8
6.80884
I ■ 73.30
0.30x621
0.023392
. 152349
7.32773
1 . 7612
0.286714
0.019697
. I4.';824
7.88079
I 7863
0.272447
0.016524
. 139565
8.46950
I . 8088
0.258793
0.013806
. 133572
9 09.53 2
1.82K6
0.2^5724
0.011480
.127844
9.7.5070
I .8461
0.233215
0.009494
.122375
10.4642
1.86x3
0. 221240
0 . 007803
. 117160
11.2101
X .8746
0.209775
0.006366
. 112194
11.0901
r.886o
0,198794
0.005149
. 107466
12 8324
1 .8957
0. 188274
0.004I2I
. 102970
13.7116
I f>039
0. 178x92
0.003257
. 098696
14.6380
I .9108
0. 168527
0.002534
. 094636
15.6130
I .9164
0. 159258
0.001933
, 000781
.087120
16.6378
X .9200
0. 150365
0.001437
17.7137
I .9245
0. 141828
0.001033
. 083646
18.8419
I .9272
0. 133630
0 . 000707
.080350
20.0235
I .9292
0. 125752
0.000450
.077224
21.2594
I . 9306
0. 118179
0.000254
.074258
22.5505
I 031.5
0. 110895
0.000112
.071447
23.8874
1 9,i 1 9
0. 103885
0,000023
0.068782
25,3006
1 .9321
Vi = 5 3571 .
- o.l \ — = 1. 9321,
~ 0.06732s : Po/p =■ 26.203 ,
APPENDIX
493
TABLE 36
3 '^
V
0
p/Po
- 0 '
Po /' piyi )
—17V'
0
I
I
0
I
0
0 . I
0-998385
0 . 995066
0.032243
1.00297
0.00032
0.2
o. 993 S 7 t
0 . 98044s
.063915
I .oijgi
0.00256
0.3
0.985640
0.956674
■094473
1,02691
0.00850
0.4
0.974730
0.924600
.123419
I , 04808
0.0197s
0-5
0,961024
0.885322
■150324
1.07562
0-03758
0.6
0.944745
0.840119
.174838
I . 10977
0 . 06294
0.7
0.926145
0 . 790366
. 196703
I . 15081
0.09638
0.8
0 . 905498
0.737456
•215751
I . 19910
0 . 13808
0.9
0 . 88309 I
0.682736
.231905
1.25502
0.18784
1 .0
00
0
0.627441
.245168
1.31903
0.24517
I . I
0.8341 5 [
0.572660
.255616
1.39163
0 . 30930
1.2
0.808180
0.519309
. 263383
1-47337
0.37927
1.3
0.781558
0.468121
. 268650
I . 56486
0.45402
1-4
0.754526
0.419650
. 271626
I . 66676
0.53239
1-5
0.727302
0.374279
.272546
I ■ 77979
0.61323
1.6
0 . 700078
0.33224s
. 271650
I. 90471
0.69542
1-7
0.673024
0.293653
. 269179
2.04233
0.77793
1.8
0.646287
0. 258507
■ 265367
2 . 19353
0.85979
1 0
0.61(7988
0. 226726
. 260438
2.35921
0.94018
;3 0
0 5(74229
0 . 1 {)8 1 7 1
. 2 54 c;q 6
2 . 5 A 017
I 0184
2 . r
0 . 569093
0. 172658
. 248029
2.73801
I . 0938
■1 'i
0 . 544642
0. 14(7976
. 240906
2.95321
I . 1660
2.3
0.520(725
0. 1 2(7901
■ 233374
3. 18708
I . 2345
2.4
0.497977
0. 1 1 2204
.225564
3 . 44.08 1
1 . 2992
2.5
0.475818
0.096657
■217585
3.71560
1.3599
2.6
0.454462
0 . 083043
■ 209533
4.01272
I .4164
2.7
0.43391 r
0.071 155
. 201486
4 33348
I .4688
2.8
0.414162
0.060801
■ 193509
4.67923
1.5171
2.9
0.395206
0.051 804
.185655
5-05138
1.5614
30
0.377026
0 . 044004
.177965
5-45134
1.6017
31
0.359606
0.037255
. I 70474
5 . 88of) i
1.6383
3-2
0.342(724
0.031428
. 163205
6.34067
1.6712
3-3
0.326(757
0 . 026406
. 156176
6 . 83309
I . 7008
3-4
0 . 3 1 1 680
0.022086
. 1 4940 1
7-35941
I. 7271
3-5
0. 297068
0.018379
. 142887
7.92125
I ■ 7504
3.6
0 . 2830(74
0.015203
. 136638
8.52020
I . 7708
3.7
0.269732
0 . 0 1 2490
■ 130655
g. 15788
1.7887
3 .«
0.256954
0.010178
. i 24(;35
9-83594
1.8041
3-9
0.244736
0.008214
.119476
10.5560
1.8172
4.0
0 . 233050
0.006551
.114273
11.3197
1 .8284
41
0.221873
0.0051 51
■109318
1 2 . 1 286
1.8376
4-2
0 . 2 1 j 1 79
0.003976
. 104604
12.9842
1.8452
4-3
0 . 200944
0.002999
.100124
13.8882
1 .8513
4-4
0 . 1 (7 1 1 46
0.002192
.095870
14.8419
I .8560
4-5
0 . 1 8 1 763
0.001535
.091832
15.8467
T . 8s(j6
4-6
0.172773
0.001008
, 088002
16.9038
1 .8621
4.7
0. 164156
0 . 000597
.084372
18.0143
I .8638
4-8
0. 1 55892
0.000291
.0S0934
19.1791
1 .8647
4-9
0. 147963
0 . 000085
0.077681
20 . 3984
I .8651
’ni = 4.9857 .
<^(771) = 0.14 142 ; = 1.8652 ,
— ^'(771) « 0.07504 ; Po/p = 21.486 .
494
STUDY OF STELLAR STRUCTURE
TABLE 37
I
-T = 0.05
3 ^
V
0
p/Po
Po /?(»?)
0
I
I
0
I
0
0. 1
0-998459
0.995141
0.030775
1.00293
0.00031
0.2
0-995863
0 . 980742
.061014
I.OII73
0 . 00244
0-3
0.986291
0.957315
, 090205
I .02649
0.00812
0-4
0.975873
0.925680
.117878
I 04735
0.01886
0.5
0.962780
0 , 886897
. 143628
1.07447
0.03591
0.6
0.947222
0.842203
. 167122
1 . 1081 1
0.06016
0.7
0.929439
0.792933
. 188111
I . 14854
0.09217
0.8
0 . 909689
0 . 740447
.206432
I . 19613
0. 13212
0.9
0.888244
0 . 686059
.222005
1.25125
0. 17982
1 .0
0.865380
0 . 630987
■2.5482s
1-314.57
0.23483
1. 1
0.841369
0.576309
■ 244958
I .38600
0 . 29640
1.2
0.816474
0.522939
.252524
1 .46670
0 . 36363
1-3
0. 790944
0.471615
.257687
1-55709
0.43540
1-4
0. 765010
0.422900
. 260643
1-65785
0. 51086
i-S
0. 738882
0.377196
.261608
T . 76972
0. 5S862
1.6
0.712747
0.334753
. 260809
1.89348
0.66767
1-7
0.686771
0.295699
.258474
2 . 03000
0 . 74699
1.8
0.661096
0. 260053
■ 254825
2 . 18019
0.82563
1-9
0-635843
0.227753
.250074
2-34.504
0.00277
2.0
0.6111H
0. 198673
.244418
2.52557
0.97767
2.1
0.586983
0. 172644
. 238040
2.72291
1 . 04<)8
2-2
0.563522
0. 149465
.231100
2 93823
1 .1185
2.8
0.540777
0. 128920
.223746
3.17276
I . 1836
2.4
0.518783
0. I 10786
.216102
3.42781
I . 2447
2.5
0.497563
0.094840
. 208279
3 - 70475
I .3017
2.6
0.477130
0.080867
. 200369
4 . 00503
1 -3545
2.7
0.457489
0 . 068663
. 192452
4.33017
I 4030
2.8
0.438637
0.058035
. i ?^4593
4.68173
1 .4472
2.9
0.420567
0.048807
.176845
5-06137
I 4873
3-0
0.403263
0.040817
. 169252
5.47081
1 -5233
31
0.386710
0.033919
. 161847
5.91181
1 - 55.54
3-2
0.3708186
0.027980
•i 5465 «
6.38621
1 -5837
3-3
0.355770
0.022884
. 147702
6 . 89590
I .6085
3-4
0.341338
0.018525
• 140995
7.44284
I . 62 ()()
3-5
o. 327.')63
0.CI4812
• 134548
8.02889
I .(>482
3.6
0.314419
0.011664
, 128363
8.65616
I . 6636
3-7
0.301881
0.009009
. 122446
9-32659
I .6763
3-8
0. 28992 [
0.006787
.116794
10.0421
I . 6865
3-9
0.278513
0 . 004944
. 111407
10.8048
1 .6045
4-0
0 . 26763 [
0.003435
. 106282
11. 6 1 62
I ■ 7005
41
0.257248
0.002222
. 101414
12.4782
1 . 704,5
4-2
0.247340
0.001276
.096798
13-3920
1 ■ 7075
4-3 1
0. 237881
0.000577
.092430
l 4 -. 5 . 5 ««
1 . 70(10
4-4
0.228846
0.000125
0 . 088306
iS. 37 Sy
1 . 70()6
rji =s 4.4601 .
= 0.22361; = I ■ 7oq6 ,
— = 0.08594 ; Po/p — 16.018 .
APPENDIX
495
TABLE 38
I
V
0
p/po
-0'
Po /?(»»)
0
I
I
0
I
0
O.I
0- 998570
0.995270
0 . 028380
I .00285
O.OOO2S
0.2
0.994340
0.978615
.056278
I .01142
0.00225
03
0-987355
0.958409
-083235
1.02579
0.00749
0-4
0-977739
0.927526
. 108826
I .04609
0. 01 741
o.S
0.965647
0.889596
.132683
I .07250
0.03317
0.6
0.951270
0.845787
. 154500
1.10526
0.05562
0.7
0-934823
0.797368
.174047
I . 1446s
0.08528
0.8
0.916542
0.745636
. 191166
I .19102
0.12235
0.9
0.896674
0.601858
.205776
I . 24478
0.16668
1 .0
0. 87547 L
0.637216
.217861
I . 30636
0.21786
I . I
0.853184
0.582768
. 227469
1-37630
0.27524
1.2
0.830056
0.529422
.234701
1-45516
0.33797
1-3
0.806319
0.477922
-239694
1-54358
0.40508
1.4
0.782186
0.428800
.242621
1 .64226
0-47554
1.5
0 - 757^^57
0 .382631
.243672
1.75198
0.54826
1.6
0.733508
0.33954s
. 243050
1.87356
0.62221
1.7
0 . 709 296
0 . 299748
. 240960
2.00792
0.69637
r.8
0 . 685358
0.263292
.237606
2.15605
0.76984
1 .0
0.661810
0.330143
.233182
2.31900
0.84179
2.0
0-93.S7S1
0,200202
.227874
2.49792
0.91149
2.1
0.616260
0, 173321
.221849
3 . 69405
0.9783s
2.2
0 . 594400
0. 149320
.215262
2 . 90869
I .0419
-’.3
0.573221
0. 127997
. 20S252
3 14327
I .1017
2.4
0.552760
0. 109142
. 200940
3 39929
I -1574
2.5
05.^^040
0.092540
• 1934.52
3.67856
I . 2090
2.6
0.514077
0,077982
. 185821
3.98218
I . 2562
2-7
0.495876
o.o()5266
.178184
4.31257
I . 2990
2.8
0.478438
0.054204
. 1 70588
4.67144
1-3374
2.6
0.461756
0 . 0446 1 8
. 163087
5.0^)081
1.3716
30
0.445816
0 . 036346
•>55726
5 . 48280
1-4015
3 • »
0 . 430604
0. 029240
.148541
5 • 93960
1.427s
3-2
0.416101
0.023167
. 141561
6.43352
1.4496
3-3
0.402285
0.018008
.134809
6.96689
I .4681
3-4
0 . 38(j 1 3 1
0.013658
.128301
7-54210
1.4832
3-5
0.376616
0.010023
. 122050
8. 16154
I -4951
3 -^>
0.364712
0.007026
. 1 1 6066
8.82752
I . 5042
3-7
0.353394
0 . 004598
• > 10355
9.54224
1.5108
3 .«
0.342632
0.0026.87
, 104922
10.3076
1-5151
3-6
0 . 33 2400
0.001258
.099771
11. 1251
1.5175
40
0.322668
0 . 000309
0 . o()49o6
11.9952
1-5185
771 = 4 . 0690 .
= o.o<)i72 ; po/p “ 12.626 .
496
STUDY OF STELLAR STRUCTURE
TABLE 39
I
0
p/po
-0'
Po/p(v)
— ij*0'
0
1
I
0
1
0
0. 1
0.998809
0.995540
0.023788
1 .00269
0.00024
0.2
0.995255
0.982302
.047195
I .01077
0.00189
0.3
0-989395
0 . 960704
.069856
1.02432
0 . 00629
0.4
0.981320
0.931414
.091432
1.04346
0.01463
0.5
0.971155
0.895308
.111625
1.06837
0.02791
0.6
0.959050
0.853418
. 130183
1 .09928
0 . 04687
0.7
0.94-5179
0 . 806876
.146911
I • 13647
0.07199
0.8
0.929733
0.756857
.161668
I . 18026
0. 10347
0.9
0.912914
0.704522
.174371
I. 23107
0.14124
1 .0
0.894929
0.650977
.184992
1.28932
0 . 1 8490
I . I
0.875985
0.597228
•I 93 S 40
1-35555
0.23419
1.2
0.856286
0.544165
.200106
1.43033
0. 28S15
1.3
0.836027
0.492537
.204759
1.51430
0 . 34604
1-4
0.815393
0.442952
. 207635
1.60821
0 . 40696
i.S
0.794554
0.393876
. 208876
1.71284
0.46997
1.6
0.773667
0.351647
. 208641
1.82908
0.53412
1.7
0.752870
0.310479
. 207094
1.95792
0 . 59850
1.8
0.732286
0.272487
. 204400
2. 1004 1
0.66226
1-9
0.712023
0.237696
. 200720
. 2.25775
0.72460
2.0
0.692170
0 . 206064
. 196210
2 . 43 I 2 I
0 , 78484
2.1
0.672804
0.177493
.191015
2.62220
0.84237
2.2
0.65398s
0,151845
.185270
2.83225
0 . 8()()7 1
2.3
0.635764
0.128954
. 1 79098
3.06302
0 . 947 .U
2.4
0.618176
0.108637
.172612
3-31631
0 .t)() 42.1
2-5
0.601249
0 . 090703
. 165908
3 • 59407
1 .0369
2.6
0.584999
0.074959
.159074
3 . 89842
I 0753
2.7
0.569436
0.061213
.152185
4.231 60
I 10(14
2.8
0.554561
0.049284
. 145308
4 59603
I . I 3()2
2.9
0.540372
0.039000
•138497
4 99426
I . 164S
30
0.526858
0.030199
. 131800
5 . 42898
I . 1862
3.1
0.514007
0.022735
.125258
5 . Q0298
1 2037
3.2
0.501800
0.016478
.118902
6.4191 1
I .2176
3.3
0.490219
0.011313
.112762
6.98018
I . 22S0
3-4
0.479240
0.007143
. 106860
7 . 58886
« 2353
3-5
0.468838
0.003897
. 101218
8.24749
1 23 <)()
3.6
0.458987
0.001540
.095856
8.9576S
1 .2423
3.7
0.449657
0.000143
0 . 090796
9.71958
1 2430
Vi = 3.7271 .
= 0.44721 ; -T 7 i</>'W = I • 2430 ,
- 0 '( 77 i) = 0.08948 ; Po/p = 9.9348 .
APPENDIX
497
TABLE 40
= 0.3
o. .
o. I
0.2
0.3
0.4
0-5
0.6,
0.7.
0.8.
0. 9.
1 .0.
1 . I .
1.2.
1.3.
1.4.
i.S-
1.6.
1.7.
1 . 8 .
1.9.
2.1
2.3
2.4
2- 5
2.6.
2.7.
2.8.
2.9.
30.
31.
3.2.
3.3.
3- 4.
35.
P/Po
I
0.999025
0.996115
0.991313
0 . 984690
0.976341
0 . 966384
• 0.954954
0.942199
0.928281
0.913362
0.897612
0.881194
0.864269
0 . 846990
0.829500
0.811932
0,794405
0.777028
0.759804
0 . 743085
0.726671
0.710710
0,695248
0 . 6803 2 2
0 . 665958
0.652177
0.638988
0.626398
0.614404
0 . 60300 r
0.592178
0.581920
0.572210
0,563026
0. 55434^1
1
0.995827
0.983429
0.963163
0.9.35601
0.901498
0.861752
0.817358
0.769361
0.718812
0.666724
0.614046
0.561631
0 . 5 10234
0 . 460436
0.412780
0 ■ 367636
0.325276
0 . 285874
0.249518
0.216226
0-iJi5<;52
0. 158610
0.134074
0. 112198
0.092818
0.075763
0 . 060860
0.047938
0 . 036833
0.027393
0.019478
0.012964
0.007754
0.003784
0.001065
-0'
o
0.019473
•038655
•057264
.075038
.091742
. 107178
.121183
• 133637
. 144462
■153622
.161115
. 166976
. 171267
.174074
•i755ot
.175664
. 174689
.172705
.169839
, 1 662 1 7
. 161961
.«S7>85
•t5i9Q4
. 146487
. 140750
. 134864
.128898
. 1 22916
. 1 16971
. X r 1 1 j I
•105377
.099806
.094430
.089278
.084376
Vx = 3.5803 .
Mv-) = O.S 4772 ; -vl4>'(Vi) =
= 0.080O4 ; po/p ■=
Po/^iv)
I .00262
I .01007
1.03256
I .04066
I .06396
I .09288
1.12767
I . 16866
I . 21622
1.27079
I ■,3328s
I .40299
I .48182
1.57007
I . 66854
1.77812
I . 89980
2.03467
2.18395
2 . 34898
2.53124
2.73236
2.95411
3 . 19844
3.46751
3.76361
4.08924
4.44710
4 . 84001
5.27096
5-74302
6.25917
6.82226
7.43466
8.09790
0.00019
0.00155
0.00515
0.01201
0.02294
0.03858
0.05938
0.08553
O.II70I
0.15362
0.19495
o . 24044
o . 28944
0.341 19
0.39488
0.44970
0.50485
0.55956
0.61312
o . 66487
0.71425
0.76078
0.80405
0.84376
0.87969
0.91168
0.93967
o . 96366
0.98372
I .00000
I .0127
1.0220
1.0283
I .0321
I . 0336
1-0337,
8.6673 .
498
STUDY OF STELLAR STRUCTURE
TABLE 41
I
V
0
p/Po
-*0'
Po / piv )
—17*0'
0
I
I
0
I
0
O.I
0.999226
0.996135
0.015456
1.00232
0.00015
0.2
0.996916
0.984643
. 030698
1.00932
.00123
0-3
0.993101
0.965821
•045517
1.02106
.00410
0.4
0.987833
0.940149
.059720
1.03763
.00956
0.5
0.981183
0.908264
.073130
1.05920
.01828
0.6
0.973239
0.877289
■085594
1 . 08596
. 0308 1
0.7
0.964100
0 . 829008
.096985
1.11815
-04752
0.8
0.953881
0.783404
.107204
I . 15607
.06861
0.9
0.942701
0.735049 .
.116184
I . 20006
.0941 1
1 .0
0.930687
0.684858
.123883
I • 25053
. 1 238S
1 .1
0.917967
0.633701
. 130290
1.30794
■15765
1.2
0.904671
0.582382
•I35417
1.37282
• 19500
1*3
0.890925
0.531620
.139299
1-44577
-23542
1.4
0.876851
0.482036
.141991
1-52747
. 27830
1.5
0.862565
0.434ISS
• 143562
1.61867
.32301
1.6
0.848174
0.388394
. 144092
I . 72023
.36888
1*7
0.833778
0.345077
.143671
I - 83300
.41521
1-8 ,
0.819468
0.304435
. 14269s
1.95832
.46136
1.9
0.805324
0.266616
. 140358
2 . 097 1 1
. 506()()
2.0
0.791418
0.231699
. 137660
2.25075
• 55064
2.1
0.777811
0.199698
.134394
2.42072
.59268
2.2
0.764555
0.170579
■ 130653
2.60861
■63236
2.3
0.751694
0.144268
. 126523
2.81621
■ 6693 1
2.4
0.739261
0.120659
. 122085
3 04547
.70321
2.5
0.727284
0.099625
.117415
3-20855
• 73384
2.6
0.715783
0.081023
.112581
3 •57778
.76105
2.7
0.704772
0 . 064704
. 107646
3 ■88573
■ 78474
2.8
0.694256
0.050515
. 102666
4.22517
. 80490
2-9
0.684238
0.038307
.097690
4-59888
.82158
30
0.674716
0.027937
.092766
5.01002
.83489
3.1
0.665682
0.019274
.087931
5 .46166
. 84502
3*2
0.657126
0.012210
.083223
5 ■ 0568 1
.85.^20
3-3
0.649032
0 . 006662
.078674
6 . 498 1 5
85676
3-4
0.641384
0.002610
.074316
7.08766
8590 <)
3-5 j
0.634161
0.000216
0.070816
7-72543
0,85978
rji = 3 5245 .
= 0.63246; —vWM = o.85g.S ,
= o.o 6 q 22 ; p„/j3 = 7.SS86 .
APPENDIX
499
TABLE 42
I
V
p/Pii
-0'
Po/lSiv)
0
0. 1
1
0.999411
I
0.996471
0.985967
0.968730
0
I
0
0.2
0.011760
1 .00212
0.0001 18
0.3
u . yy 7053
0.994747
0.990730
0.985650
0.979571
■023371
1.00851
.000935
.003122
■007292
.013976
.023605
.036491
.052824
.072665
.095952
.122506
. 152045
.184201
•218533
• 254550
•291732
■329538
•367432
.404898
•441445
.476626
.510038
-.541335
.570227
.596485
.619941
. 640487
.658076
.672721
. 684490
.693513
•699979
. 704141
■ 706344
0.707033
0.4
. 034689
I .01922
0. i;
0.945149
■ 04 SS 7 S
I- 03434
0.6
0.915750
0.881163
0.842106
. 055906
I. 05401
0.7
.065570
1.07841
0.8
^ ‘ y 7*=502
.074472
I. 10774
0.0
0.799351
0.753699
■082S38
I . 14227
i.o
0.946793
.089710
1.18232
I . I
0.705949
0.656883
0.607237
0.557680
0.508825
0.461195
.095952
1.22822
1.2
0.926576
0-915839
.101245
1 . 28043
1.3
•105587
!■ 33938
1.4
. 108995
1.40563
1.5
0.803569
0.882208
0.870802
0.850425
0.848142
0.837012
0.826086
0.815410
0 . 805022
0-704953
0.785230
. I I 1496
1.47979
1.6
•II3I33
1-56255
1.7
0.415233
0.371298
0 .329672
0 . 290560
• 113958
1.65467
1.8
. 114027
I. 75701
I . t)
•^>3405
1-87057
2.0
.112160
1.99641
2 . r
1 0.254(00
0.220367
0 . j 89387
0 . 1 60974
0.135569
. 110361
2.13573
2 2
.J 080 78
2 . 28989
2.3
. 105380
2 .46036
2.4
. 102332
2.64881
2 . s
. 098998
2.85706
2.1)
0 . I I 2593
■095438
3 08713
2.7
0 . 766892
0.092107
. o() 1 707
3.34121
2.8
0. 073992
0 . 058 I 20
0.044362
.087858
3.62172
2 .g
0-750105
0.74 -’303
0 *7 j ,1 1
■ 083938
3.93126
3.0
.079991
4 . 27261
3.1
0.032590
0. 022?)85
.076054
4 . 64868
3.2
/i 54 ‘^y 3
0. 727868
.0721 66
5.06249
3.3
0-014544
0 . 008093
.068357
5.51695
3.4
. / «: 1 J 1 0
0.714931
• 064659
6.01473
3 . s
0 . 0033 • y
. 061 102
6-55774
Vii^ . /vjnxjK) j
0 . 000390
0.057717
7 - 14658
= 3 • 5330 .
< I>M = 0.707.07 ; = 0.70704 ,
-4- (>;.) = o . 05O644 ; A./i5 = 7 . ISOS .
STUDY OF STELLAR STRUCTURE
Soo
TABLE 43
V
0
p/Po
-0'
Po/'pM
“> 7 * 0 '
0
I
I
0
I
0
0. 1
O.Q 99 S 788
0.996843
0 . 0084087
1.00285
0 . 000084
0.2
0.9983198
0-987436
.0167381
1.00761
.000670
0-3
0.9962374
0.971967
. 0248708
1.01719
.002238
0.4
0 . 9933 S 49
0.950741
.0327264
I . 03070
.005236
O-S
0.9897042
0.924I7I
.0402225
1.04826
.010056
0.6
0.9853249
0.892758
.0472850
I . 07003
.017023
0.7
0.9802638
0.857079
.0538498
1.09618
. 026386
0.8
0-9745733
0.817762
.0598630
I . 12694
. 0383 1 2
09
0.9683110
0.775468
.0652821
I . 16256
.052878
1 .0
0961S377
0.730873
.0700764
I . 20336
.070076
I . I
0.9543172
0.684647
.0742266
I . 24969
. 0898 14
1.2
0.9467142
0.637436
.0777247
1.30194
. III924
1-3
0-9387939
0.589852
.0805729
1.36058
. 136168
1.4
0.9306209
0.542460
.0827830
1.42612
.t 622 SS
1.5
0.9222579
0-495766
.0843752
1.49915
. 189844
1.6
0-9137655
0.450217
-0853772
1-58033
. 218566
1.7
0.9052010
0.406193
.0858223
1.67039
. 248026
1.8
0.8966183
0.364011
.0857490
1.77016
.277SJ7
1-9
0.8880672
0.323922
.0851990
I . 88056
- 507 S <>8
2.0
0.8795930
0.286120
.0842168
2.00263
,336867
2.1
0.8712367
0.250740
.0828478
2.13750
-. 56 S 3 . 5 <)
2.2
0.8630348
0.217868
.0811382
2 . 28647
-362701)
2.3
0.8550189
0.187543
.0791338
2.45095
.418618
2.4
0.8472163
0.159727
.0768970
2.63191
.442927
2-5
0.8396500
0.134514
•074417s
2.83292
.465109
2.6
0.8323383
0.III723
.07179x0
3.05402
•485307
2-7
0.8252960
0.091316
.0690381
3 • 29795
.503288
2.8
0.8185337
0.073204
.0661952
3.56698
.518970
2.9
0.8120588
0.057283
.0632962
3.86357
•532321
30
0.8058753
0.043447
.0603729
4.19033
•.5433.56
31
0 . 7999840
0.031593
.0574542
4.54997
•5.5213.5
3.2
0. 7943834
0.021622
.0545666
4-94529
.558762
3-3
0.7890688
0.013456
•0517354
5.37892
• 563398
3.4
0 . 7840336
0.007051
.0489844
.5.85315
. 5662(10
35
0.7792685
0.002445
•0463373
6.36951
.567632
3-6
0.7747620
0 . 00001 6
0.0438166
6.92839
0.567863
97i = 3 . 6038 .
<l>(vt) = o,774SQ7 ; “ 0.56786 ,
— tpXrii ) = 0.043724 ; Po/p = 6.9504 .
APPENDIX
SOI
TABLE 44
V
0
p / Pq
-r
Po/pM
0
I
r
0
I
0
0. 1
0.9998510
0.99777
0.0029774
I .00134
0 . 000030
0.2
0.9994053
0.99110
•0059309
I .00538
.000237
0-3
0 . 9986664
0 . 98008
.0088370
I .01217
. 000795
0.4
0.9976402
0.96485
.0116730
I .02165
.001868
0-S
0.9963349
0-94563
.0144170
I . 03400
.003604
0.6
0.9947606
0.92265
. OT 70489
1.04925
.006138
0.7
0.992929s
0 . 89620
.0195501
1.06751
. 009580
0.8
0.9908555
0 . 86662
.0219039
I .08891
.014018
0-9
0.9885541
0.8342s
.0240960
1.11358
.019518
1.0
0.9860421
0.79947
.0261144
1.14168
.026114
i.i
0.9833.574
0,76268
.0279490
1.17341
.033818
1.2
0.9804587
0.72427
.0295923
z . 20900
.042613
1.3
0.9774254
0.68465
.0310398
I . 24867
.052457
1-4
0.9742574
0.64418
.0322885
I . 29272
.063285
1-5
0.9709744
0.60326
.0.5.53.582
I. 34144
.075011
1.6
0.9675963
0.56224
.0341907
1.39520
.087528
1-7
0.9641427
0.52146
.0348498
1-45436
.100716
1.8
0.9606326
0,48121
.0355210
1.51937
.114440
1-9
0.9570845
0.441 78
.03561 14
I . 59070
.128557
2.0
0.9535161
0.40341
.0357296
1.66888
. 142918
2.1
0.9490440
0.36632
.o 3.';6854
I . 75450
.157373
2.2
0.9463841
0
0
6
■0354891
1.84821
.171767
2.3
0.9428509
0 . 29669
■0351524
I. 95073
.185956
2.4
0.9393579
0. 26442
.0346870
2 . 06285
.199797
2.5
0.9359174
0. 23398
■0341053
2 . 1 8546
.213158
2.6
0.9325403
0.20543
.0334198
2.31949
.225918
2.7
0.9292364
0 . 1 7882
.0326428
2 . 46604
.237966
2.8
0.9260144
0. 15417
.031 7867
2.62625
. 249208
2.9
0.9228814
0. 13147
.0308637
2.80139
-259564
30
O.Q 108435
0. i 1071
. 0298853
2.99287
. 26S968
31
0 . 9 169058
0 . CK) 1 856
,0288630
3.20217
•277373
3-2
0.9140720
0 . 074869
.0278077
3.43090
. .284751
3-3
0.9113450
0 . 059699
.0267297
3.68081
. 291086
3-4
0.9087265
0.046289
.0256389
3.95370
. 2963S6
33
0.0062173
0.034584
•0245445
4.25146
. 300670
3-6
0.9038174
0.024533
■0234553
4-57599
.303981
3-7
0.9015258
0.016094
.0223799
4.92910
.306381
3.8
0 8993407
0.009251
.0213264
5.31239
.307955
3-9
0.8972505
0.004042
.0203034
5.72690
.308815
40
0.8952787
0 . 000665
0.0193196
6.1728s
0.309114
Vi =» 4 - 044 ^’ •
= 0.S94427 ; = 0.30912 ,
-0'(77i) = 0.018896; Po/p => 6.3814 .
GENERAL INDEX
GENERAL INDEX
Ahhauproscsse^ 468
Absorption coeflicient, igo
Rosseland mean, 212, 263
in terms of Einstein coeflicients, 191
Adiabatic changes
infinitesimal, 16
for matter and radiation, 55
for perfect gas, 39
quasi-statical, 16
Adiabatic exponents for matter and radi-
ation, 56
table of, 59
Adiabatic inclosurcs, 1 2
Alpha decay
theory of, 460
of uranium, 458
Anderson, W., 409, 422, 451
Atkinson, K. d’E., 456, 485
A iiflnw prozrssv, 498
Beer, A., 314, 32 1
Bcthe, II., 474, 484-S6
Betti, K., loi, 180
Bialobjesky, I., 229, 248
Biermann, L., 227, 248, 35O
Black-body radiation, 53
Bohr, N., 256, 470
Brock, K. K., 369, 410
Bruggencate, P. ten, 451
Capella, 75, 221, 276-77, 289, 305, 308
Carath6odory, C., ir, 23-24, 32, 34, 37,
357
Carathcodory^s principle, 24
Carath6odory’s theorem, 23
Central condensiition of stars, 292, 303-30
dependence on chemical composition,
307
systematic variation in H.R. diagram,
3157320
tables for the evaluation of, 302-4
Central pressure
minimum of, 65, 72
in stars, 230, 232, 354
Chandrasekhar, S., 72, 75, 82-83, 96, 182,
219, 24S, 292, 314', 321, 328, 356,
397, 409, 411, 415, 422, 426, 44O,
45 i» 491
Cluster diagrams, Kuiper’s inteipretation
of, 286
Companion of Sirius, 412, 414
Composite configurations, 170, 438
Condon, E. U., 459
Contraction hypothesis, 454
Convection currents, 85, 225, 483
Convective equilibrium, 84
Cosmogenetic clianges, 48
Cowling, T. G., 227, 248, 332, 333, 356,
468
Darwin, C. O., 409
Darwin, G. H., 180
Degeneracy
criterion for, 393
stellar criterion for, 434, 436, 443
Degenerate configurations, 415
approximation for small central densi-
ties, 419
eficct of radiation pressure in, 437
limiting mass, it/,, 421
observational verification of the theory
43 1
physical characteristics of, 417, 425,
427-30
Degenerate electron gas
elementary treatment, 357
equation of state of, 360-61
general treatment, 388
internal energy of, 360-61
•Dciiterons as a source of neutrons, 477
Diathermic partitions, 12
Dirac, P. A. M., 363, 366, 409-10
Dirac equation, 363, 432
solution of, 364
Disintegration of elements, 457, 468
criterion for, 470
Domain of degeneracy, 440, 442
nature of curves of constant mass in,
442, 445
Karth^s atmosphere, temperature gradi-
ent in, 85
Eddies, 225
Eddington, A. S., 51, 60, 72, 82, 215, 221,
248, 278 IT., 291, 321, 410, 456, 485
Eddington's quartic equation, 229, 243,
305
STUDY OF STELLAR STRUCTURE
506
Ehrenfest AfFanasJewa, T., 34, 37, 409
Einstein coefficients, 189, 191, 204
Einstein-Bose distribution, 382, 384
Electron gas
degenerate case, 357, 388
general formulae for, 382, 385, 387
nondegenerate case, 394, 398
specific heat of, 394
unrelativistic case, 399
unrelativistic degenerate case, 401
unrelativistic nondegenerate case, 401
Electron scattering, Thomson’s formula
for, 270
Emden, R., 40, 43, 48, 53, 60, 84, 94, 96,
176, 180-81, 248
Emden’s thermodynamical theorem, 41,
48
Emden’s transformations, 90, 157
Emden’s variables, 43
Emission coefficient, 189
in terms of Einstein coefficients, 190
Entropy, 27, 370
principle of increase of, 31
Equation of transfer, 198, 206
solution for far interior, 207
Equations of fit, 172
methods of solution of, 173
Fairclough, N., 182
Feather, N., 473, 484
Fermi, E., 409, 41 1, 470
Fermi-Dirac distribution, 382, 384
Fermi’s law of velocity distribution, 387
First law of thermodynamics, 14
Fowler, R. H., 84, 124, i8r, 257, 261, 291,
409 ffi, 451
Fowler’s theorem, 124 ff., 136, 150
extension to cases « < i, 144 ff.
Free energy, 35, 372
Frenkel, J., 409
Gamow, G., 459-60, 472, 484-86
Gamow factor, 457, 463
Gaposchkin, S., 314
Gaunt, J. A., 262, 283
Gaunt factor, 262-63
Gibb’s ensemble, 369 ff.
Gravitational equilibrium, the equations
of, 62, 213, 225
Gravitational potential, 63
Green, G., 95, i8o
Guggenheim, E. A., 261, 291
Guillotine factor, 250
table of, 269
variation through a star, 275
Gurney, R. W., 459
Hardy, G. H., 123
Hardy’s theorem, 123
Harkins, W. D., 456, 485
Helium content of stars, 255, 287, 482
Helmholtz, H. von, 453-55, 484
Hertzsprung, E., 287, 291
Hertzsprung gap, 285
Hertzsprung-Russell diagram, 251
Strdmgren’s interpretation of, 280 ff.
Hilbert, D., 215
Hill, G. W., 180
Homer Lane’s function, 88
Homologous family of solutions, 103
Homologous transformations, 8r, 233
Homology-invariant functions, 101 ff.
159 ff-
Hopf, E., 115, 1 19, 181
Houtermans, F. G., 456, 485
Hund, F., 409-10
Hyades cluster
central condensations of stars in, 3 1 2
stars in, 287
Hydrogen content, 255
curves of constant
in (mass, radius) diagram, 283
in H.R. diagram, 285
spreading-out of, 282, 290
of stars, 276, 287, 433, 489
Hydrogen and helium content of stars
288-89
Instability for radial oscillations, 52
Integral theorems
on the equilibrium of a star, 61 ff.
on the radiative equilibrium of a star,
216 ff.
Integrating denominator, 19
Isothermal equation, 156
homology theorem for, 1 58
reduction to first order, 159
singular solution of, 157
in (//, v) plane, 160
discussion of, t08
in (y, s) plane, discussion of, lOt
Isothermal E-solutions
starting series for, 1 50
uniqueness of, lOi
GENERAL INDEX
507
Isothermal functions
asymptotic behavior of, 164, 167
oscillatory character about the singular
solution, 166
Isothermal gas sphere, 155, 447
Jeans, J. H., 52, 295, 321, 356
Joliot, F., 486, 488
Joule-Kelvin experiment, 30
Juttner, F., 387, 394-g5. 410
Keenan, V. C., 350
Kelvin, Lord, 24, 30, 34, 40, 84-85, 88, 95,
I76-79> 453-55, 4^4
Kelvin’s transformation, 8q, 157
Kirchhoff’s laws, 199 ff.
Kothari, D. S., 411
Kramers, H. A., 262, 283
Kuiper, G. P., 2, 287, 291, 310 n., 415,
43L 452
Landau, L., 409
Lane, H., 47, Oo, 484-85, 88, 176-78
Lane-Kmdcn equation, 88
asymptotic behavior of the solutions of,
121, 123, 134, 13O, 143
discussion of, in (//, v) plane, 146
arrangement of solutions, 152
discussion of, in (y, s) plane, 107-39
arrangement of solutions, 126, 130,
142
spiraling about the second singular
point, 134, 143
/7-solutions of, 142
equivalent iirst-order differential equa-
tion, 103, 106
AJ-solutions of, 103, 106, iig
/^’-solutions of, 120
homologous family of solutions of, 103
homology theorem for, 101
^/-solutions of, 1 20
0-solutions of, 144
singular solution of, 89
transformations of, 8g
Lane-l^mdcn function, 88
constants of, 96
for general 94
for n = o, 91
for « = 1, 92
for n = 5, 93
starting series for, 95
I.ane’s theorem, 47
Light quanta, statistics of, 405
Local thermodynamical equilibrium, 205
Luminosity formula, 218, 290
for model €= constant, 326
for point-source model with «= con-
stant, 350
for point-source model with
355
for standard model, 232
for stars with negligible radiation pres-
sure, 234, 238
for stars with varying kt], 243-45
use of, 249, 251
HJfi, 437, 440, 442, 445
1^3,421,441,443-45
Majumdar, R. C., 410-11
Mass-luminosity diagram, i, 3
Mass-radius diagram, i, 4, 281-82
Maxwell’s law of distribution of veloci-
ties, 263, 387
Mean molecular weight, 254
of degenerate electron gas, 432
first approximation for perfect gas, 255
of Russell mixture, 260
second approximation for perfect gas,
256
of stars
determination of, 272
existence of two solutions, 277, 309
Mean pressure, minimum of, 70, 78
Mean temperature, minimum of, 68, 76
Miller, J. C. P., 95
Milne, K. A., 64, 72-73, 82-83, H7, 182,
214-15, 248, 32r, 356, 438, 451, 457
Molecular weight. See Mean molecular
weight
Miller, C., 409
Negative density gradients, 331
Neumann, J. von, 332 IT., 356, 409
Neutrons, transmutations by the capture
of, 470, 478 IT.
Newcomb, S., 177
Nondegenerate electron giis, 394
general formulae, 396
internal energy of, 397
specific heat of, 397-98
Nuclear reactions, 474-”78
as autocatalytic chains, 474-76
model chain of, 478
Nuclei, light
reaction chains involving, 474 fl., 486
stability criterion for, 472
Nucleus of mass 5, 474, 480
STUDY OF STELIAR STRUCTURE
So8
Opacity coe£&cient, 212
average through a star, 217
maximum of, 219
including scattering and ionization, 271
maximum of, 222
Rosseland mean, 212, 263
for Russell mixture, 269
stellar, 261
Opacity discrepancy, 278
Parameters of a star, i
Partially degenerate configurations, 446-
49
Pauli, W., 371, 376, 410-11
Pauli principle, 358, 384
Payne-Gaposchkin, C., 272
Peierls, R., 369, 410
Penetration of nuclear barriers, 464 ff.
Perrin, J., 456, 485
Perry, J., 70
Perturbation theory
for models with varying kti ,. 239
for point-source model, 344, 346
Pfafiian differential equation, mathe-
matical theorems on, 17
Physical and astronomical constants,
487-88
Planck, M., 395, 410
Planck’s law, 204, 384, 408
Poincar6, H., 51, 53, 60
Polytropes
complete, 88
physical characteristics of, 97-101
composite, 170
theorem on, 1 75
Polytropic change, 40
equation for, 41
Polytropic equilibrium, 84
Pol3rtropic index, 44
effective, 324, 351
Potential barriers, transparency of, 463 ff .
Potential energy (gravitational), 63
minimum of, 68, 76
of pol3rtropes, loi
Problem of stellar structure, 2
Protons, transmutations by the capture
of, 470 ff .
Quantity of heat, 14
Quantum states of an electron gas, 358
enumeration of, 363, 367
Radiation
definitions, 182, 184, 187, 19 1, 194
mechanical force exerted by, 196^8
pressure of, 192
Radiation field, normal modes of, 402
Radiation pressure in stars, 62, 213 ff.
according to Eddington’s quartic equa-
tion, 229, 232, 305, 489
maximum of
central, 74
table of, 75
Radiative equilibrium
equation of 207
with convection, 227
stability conditions for, 222, 224, 237
Ritter, A., 47, 53, 60, 66, 68, 70, 82, 84-85,
loi, 176-79
Ritter’s relation, 70
Rosseland, S., 207, 212, 215, 291, 356
Rudzki, P., 47, 180
Russell, H. N., 182, 251, 272, 291, 356
Russell mixture, 259, 266
hydrogen and, hypothesis, 272
hydrogen, helium, and, hypothesis, 289
mean molecular weight of, 260
opacity coefficient for, 269
Sadler, D. H., 95
Schuster, A., 94, iSo
Schuster- Emden integral, 94, 136, 154
Schwarzschild, K., 86, 181, 206, 215, 224,
248
Second law of thermodynamics, 24, 29
See, T. J. J., 180
Severny, A. B., 356
Siedentopf, H., 248, 415, 451
Sommerfeld, A., 410-11
Sommerfeld’s lemma, 389, 401
Specific heats
of a degenerate electron gas, 394
of matter and radiation, 59
of a nondcgcncratc electron gas, 397-98
of a perfect gas, 38
Standard model, 228
breakdown of, 285
minimal characteristic of, 234
perturbation theory for, 239
physical characteristics of, 229-32
Stcensholt, G., 408, 485
Stcfan-Boltzmann constant, 203, 205
Stefan-Boltzmann law, 53, 182
Stellar configurations with degenerate
cores, 438 ff.
GENERAL INDEX
509
Stellar envelopes, 292
with negligible radiation pressure, 299
Stellar models, s
with € oc r, 244
with 17 a 327
with negligible radiation pressure, 234
point source with constant, 332
solutions for, 350
. point source with Kocpr-J.s^ 351
standard, 228, 449
with uniform distribution of energy
sources, 244, 322
with varying /07, 239
Sterne, T. E., 409, 457
Stoner, E. C., 361, 409, 422, 45^-52
Stromgren, B., 218, 234, 238, 248, 251,
256, 259 ff., 269 ff., 2875., 309 ff-,
321, 356, 4 iS» 4 Si> 4 S 7 i 485
Struve O., 317, 321
Swirles, B,, 411
Synthesis
of elements, 468
criterion for, 470 ff.
of heavy elements, 470» 478
Teller, K., 472, 485
Temperat ure
absolute, 27
empirical, 12
on gas thermometer scale, 29
polytropic, 43
of stars, central, 230, 232, 326, 354, 4S9
Thermodynamical potential, 35, 380
Thiesen, M., 180
Time scale
evidence for, 455
of Helmholtz and Kelvin, 453, 455
transmutation, 456
Tolman, R. C., 410
Transmutation of elements
by charged particles, 456, 464, 472
by neutron captures, 470, 478 £E.
occurring at nonequilibrium rates, 456,
458
by proton captures, 474 ff.
as a source of stellar energy, 456
Trumpler, R. J., 287, 291, 321
Trumpler stars, 305, 490
central condensations of, 313
Tuominen, J., 356
Turbulent energy, transport of, 226
Uniform expansion and contraction
of gaseous configurations, 45, 453
of polytropic gas spheres, 48
Virial theorem, 49, 424
applications of, 51 ff.
Vogt, H., 251, 290, 321
Vogt-Russell theorem, 252, 281, 457
Weizsacker, C. F. von, 288, 468 ff.
von Weizsacker*s hypothesis, 469
Weyl, H., 404, 411
White dwarf, 412, 490
gaseous fringe of, 413
hydrogen content of, 433
Kuiper’s, 431
van Maanen’s, 433
White-dwarf functions
constants of, 426
tables of, 4Q2 ff.
Wilson, A. H., 485
Wilson, 1 C. D., 485
Zeuner, G., 40, ho
Zlotowski, J., 486, 488
ZullncT, E., i8o
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